%   This is a pure LaTeX 2e file
%   File name:   nf-ltp-lb-TeXPower.tex
%		(Normal Frames and Linear Transports along Paths in Line Bundles)
%   Last update: August 23, 2006 %   Final version:
%   Purpose:	 Scientific presentation. Report presented at the
%	8th International Workshop on Complex Structures and Vector Fields

%   Author (and responsible for all errors):
%***************************************************************************%
%**                                                                       **%
%**	   BOZHIDAR ZAKHARIEV ILIEV:				 	  **%
%**                                                                       **%
%**	   Laboratory of Mathematical Modeling in Physics		  **%
%**	   Institute for Nuclear Research and Nuclear Energy              **%
%**	   Bulgarian Academy of Sciences                                  **%
%**	   Boul. Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria         **%
%**	   E-mail address: bozho at inrne.bas.bg			  **%
%**        URL: http://theo.inrne.bas.bg/~bozho/	      		  **%
%**                                                                       **%
%**			(May be used freely by everybody)                 **%
%**                                                                       **%
%***************************************************************************%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                                                                       %%
%% 		Beginning of \CharacterTable                             %%
%%      	                                                         %%
%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z  %%
%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z  %%
%%   Digits        \0\1\2\3\4\5\6\7\8\9                                  %%
%%   Exclamation   \!     Double quote  \"     Hash (number) \#          %%
%%   Dollar        \$     Percent       \%     Ampersand     \&          %%
%%   Acute accent  \'     Left paren    \(     Right paren   \)          %%
%%   Asterisk      \*     Plus          \+     Comma         \,          %%
%%   Minus         \-     Point         \.     Solidus       \/          %%
%%   Colon         \:     Semicolon     \;     Less than     \<          %%
%%   Equals        \=     Greater than  \>     Question mark \?          %%
%%   Commercial at \@     Left bracket  \[     Backslash     \\          %%
%%   Right bracket \]     Circumflex    \^     Underscore    \_          %%
%%   Grave accent  \`     Left brace    \{     Vertical bar  \|          %%
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%%                                                                       %%
%% 		End of \CharacterTable                                   %%
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% \documentclass[12pt,titlepage,a4paper]{article} %QQQQQQQQQQQQQ

% beginning of the file nf-ltp-lb_CSVF2006_report.bbl
% ATTENTION: contains dynamic effects! Restore them if regenerating the bbl file!
    \begin{filecontents}{nf-ltp-lb_CSVF2006_report.bbl}
\begin{thebibliography}{10}

\pageDuration{9}

\bibitem{Konopleva&Popov}
N.~P. Konopleva and V.~N. Popov.
\newblock {\em Gauge fields}.
\newblock Hardwood Academic Publishers, Chur-London-New York, second edition,
  1981.
\newblock Translation from Russian: Atomizdat, Moscow, 1972 (1~ed.), 1980
  (2~ed.).

\bibitem{Drechsler&Mayer}
W.~Drechsler and M.~E. Mayer.
\newblock {\em Fibre bundle techniques in gauge theories}, volume~67 of {\em
  Lecture notes in physics}.
\newblock Springer-Verlag, Berlin-Heidelberg-New York, 1977.

\bibitem{Baez&Muniain}
John Baez and Javier~P.\ Muniain.
\newblock {\em Gauge fields, knots and gravity}, volume~4 of {\em Series in
  knots and Everything}.
\newblock World Scientific, Singapore-New Jersey-London-Hong Kong, 1994.

\bibitem{Lorenz/1867}
Ludwig~Valentin Lorenz.
\newblock {\"Uber die Intensit\"at der Schwingungen des Lichts mit den
  elektrischen Str\"omen}.
\newblock {\em Annalen der Physik und Chemie}, 131:243--263, 1867.

\bibitem{Whittaker-History}
Edmund Whittaker.
\newblock {\em A history of the theories of aether and electricity}, volume 1.
  The classical theories. of {\em Harper torchbooks / The science library}.
\newblock Harper \& brothers, New York, 1960.
\newblock Originally published by Thomas Nelson \& Son Ltd, London, 1910;
  revised and enlarged 1951. See also the 1989 edition: New York: Dover, 1989.

\newslide

\bibitem{Roman-QFT}
Paul Roman.
\newblock {\em Introduction to quantum field theory}.
\newblock John Wiley\&Sons, Inc., New York-London-Sydney-Toronto, 1969.

\bibitem{Gockeler&Schucker}
M.~{G\"ockeler} and T.~{Sch\"ucker}.
\newblock {\em Differential geometry, gauge theories, and gravity}.
\newblock Cambridge Univ.\ Press, Cambridge, 1987.

\bibitem{Aharonov&Bohm}
Y.~Aharonov and D.~Bohm.
\newblock Significance of electromagnetic potentials in quantum theory.
\newblock {\em Phys.\ Rev.}, 115(3):485, 1959.

\bibitem{Bernstein&Phillips}
H.~J. Bernstein and A.~V. Phillips.
\newblock Fibre bundles and quantum theory.
\newblock {\em Scientific {America}}, 245:94--104, July 1981.
\newblock Reprinted: Uspekhi Fizicheskikh Nauk, vol.~136, No.~4, pp.~665--692,
  1982, In Russian.

\bibitem{bp-Frames-n+point}
Bozhidar~Z. Iliev.
\newblock Normal frames and the validity of the equivalence principle: {I}.
  {Cases} in a neighborhood and at a point.
\newblock {\em Journal of Physics A: Mathematical and General},
  29(21):6895--6901, 1996.
\newblock DOI~No.~10.1088/0305-4470/29/21/020\\ http://arXiv.org e-Print
  archive, E-print No.\ gr-qc/9608019, August 1998.

\bibitem{bp-Frames-path}
Bozhidar~Z. Iliev.
\newblock Normal frames and the validity of the equivalence principle: {II}.
  {The} case along paths.
\newblock {\em Journal of Physics A: Mathematical and General},
  30(12):4327--4336, 1997.
\newblock DOI~No.10.1088/0305-4470/30/12/019\\ http://arXiv.org e-Print
  archive, E-print No.\ gr-qc/9709053, September 1997.

\bibitem{bp-Frames-general}
Bozhidar~Z. Iliev.
\newblock Normal frames and the validity of the equivalence principle: {III}.
  {The} case along smooth maps with separable points of self-intersection.
\newblock {\em Journal of Physics A: Mathematical and General},
  31(4):1287--1296, January 1998.
\newblock DOI~No.~10.1088/0305-4470/31/4/016\\ http://arXiv.org e-Print
  archive, E-print No.\ gr-qc/9805088, May 1998.

\bibitem{bp-PE-P?}
Bozhidar~Z. Iliev.
\newblock Is the principle of equivalence a principle?
\newblock {\em Journal of Geometry and Physics}, 24(3):209--222, 1998.\\
http://arXiv.org e-Print
  archive, E-print No.\ gr-qc/9806062, June 1998.

\end{thebibliography}
    \end{filecontents}
% end of the file nf-ltp-lb_CSVF2006_report.bbl


% beginning of the file bozhomac.sty
    \begin{filecontents}{bozhomac.sty}
%***************************************************************************%
%**     BEGINNING of LaTeX2e style package of BOZHIDAR ZAKHARIEV ILIEV    **%
%**	Filename: bozhomac.sty                                            **%
%**	Author (and responsible for all errors):                          **%
%***************************************************************************%
%**                                                                       **%
%**	  BOZHIDAR ZAKHARIEV ILIEV:					  **%
%**                                                                       **%
%**	  Laboratory of Mathematical Modeling in Physics		  **%
%**	  Institute for Nuclear Research and Nuclear Energy               **%
%**	  Bulgarian Academy of Sciences                                   **%
%**	  Boul. Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria          **%
%**	  E-mail address: bozho@inrne.bas.bg                              **%
%**       URL: http://theo.inrne.bas.bg/~bozho/		       		  **%
%**                                                                       **%
%**                                                                       **%
%**			(May be used freely by everybody)                 **%
%**                                                                       **%
%***************************************************************************%

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%% 		Beginning of \CharacterTable                             %%
%%                                                                       %%
%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z  %%
%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z  %%
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%% 		End of \CharacterTable                                   %%
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% ======================== identification ===================================
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
\ProvidesPackage{bozhomac}[2006/8/19
			   BOZHIDAR Z. ILIEV's LaTeX2e style package]


% ========================  initial code ====================================

% ---------> Beginning of Package Loading part

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%	Beginning of full AMS-LaTeX package
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\newtheorem{Thm}{Theorem}[section]			% [ THEOREM ]

\newtheorem{Prop}{Proposition}[section]			% [ PROPOSITION ]

\newtheorem{Lem}{Lemma}[section]     			% [ LEMMA ]

\newtheorem{Cor}{Corollary}[section] 			% [ COROLLARY ]

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% to produce variants of the theorems-like ... numbering Redefine \theVar... !!!
% \theoremstyle{plain}	% plain, definition, or remark
\newtheorem{VarThm}{Theorem}			% [THEOREM ]
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\newtheorem{VarPost}{Postulate}			% [ POSTULATE ]
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\newtheorem{VarProp}{Proposition}		% [ PROPOSITION ]
\newtheorem{VarLem}{Lemma}		     	% [ LEMMA ]
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	\theoremstyle{definition}	%*********************************
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\newtheorem{Assr}{Assertion}[section]			% [ ASSERTION ]

\newtheorem{Conc}{Conclusion}[section]			% [ CONCLUSION ]

\newtheorem{Exmp}{Example}[section]			% [ EXAMPLE ]

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\newtheorem{Prob}{Problem}[section] 			% [ PROBLEM ]
%	Versions
\newtheorem{VarDefn}{Definition}		% [ DEFINITION ]
\newtheorem{VarConj}{Conjecture} 		% [ CONJECTURE ]
\newtheorem{VarAssr}{Assertion}			% [ ASSERTION ]
\newtheorem{VarConc}{Conclusion}		% [ CONCLUSION ]
\newtheorem{VarExmp}{Example}			% [ EXAMPLE ]
\newtheorem{VarExrc}{Exercise}			% [ EXERCISE ]
\newtheorem{VarProb}{Problem} 			% [ PROBLEM ]

	\theoremstyle{remark}	%*********************************
%	\theorembodyfont{\rmfamily\upshape} % requires amstex package

\newtheorem{Rem}{Remark}[section] 			% [ REMARK ]

\newtheorem{Comm}{Comments}[section] 			% [ COMMENT ]

\newtheorem{Note}{Note}[section] 			% [ NOTE ]

\newtheorem{Case}{Case}[section] 			% [ CASE ]
%	Versions
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\newtheorem{VarComm}{Comments} 			% [ COMMENT ]
\newtheorem{VarNote}{Note} 			% [ NOTE ]
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%	\theoremstyle{remark}	%*********************************
 %	\theorembodyfont{\sffamily\upshape}  requires amstex package

\newtheorem{Algr}{Algorithm}[section] 			% [ ALGORITHM ]

\newtheorem{Claim}{Claim}[section] 			% [ CLAIM ]

\newtheorem{Ackn}{Acknowledgment}[section]		% [ ACKNOWLEDGMENT ]
%	Versions
\newtheorem{VerAlg}{Algorithm} 			% [ ALGORITHM ]
\newtheorem{VerClaim}{Claim} 			% [ CLAIM ]
\newtheorem{VerAck}{Acknowledgment}		% [ ACKNOWLEDGMENT ]

%++++++++++++++++++++++++++++++++++++++----------------------------------------
%	END of theorem-like environments (BOZHO'S DEFINITIONS):
%++++++++++++++++++++++++++++++++++++++----------------------------------------


%  >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
%  BEGINNING of New and Renew commands and environments (BOZHO'S DEFINITIONS):

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%	REDEFINE the corresponding counters and there appearance!!
\newenvironment{Version}[1][Choose a NAME and FONT!]%
		{\vspace{0.0ex}\noindent{#1}}%
		{\vspace{1.2ex}}

% {Proof}: an alternative is the AMS (amsthm package) "proof" environment
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% Caution: some packages like FoilTeX also (re)define this environtment
\newenvironment{Proof}[1][Proof.]%
		{\vspace{0.0ex}\noindent\textit{#1\hspace*{0.5em}}}%
		{\hspace*{0pt}\hfill\QEDsymbol\vspace{1.2ex}}
% Q.E.D.  symbols. The AMS package `amsthm` provides \qed producing \qedsymbol
\providecommand{\qedsymbol}{$\square$}  % Requires amssymb package
    \newcommand{\QEDsymbol}{$\square$}	% Requires amssymb package
\providecommand{\qed}{\hfill\qedsymbol}
%\newcommand{\QED}[1][6]{\hfill\hbox{\vrule width #1pt height #1pt depth 0pt}}
\newcommand{\QED}{\hspace*{0pt}\hfill Q.E.D.}

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\newcommand{\EndCor}{\hspace*{0pt}\hfill$\blacksquare$}
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%
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%
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%
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%	Mathematical Abbreviations:
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\newcommand{\const}{\mathrm{const}}	%  constant quantity
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%
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\DeclareMathOperator{\ad}{ad}	%   ... the differential of \Ad ...
% \newcommand{\exp}{\mathrm{exp}}	% exponential mapping; LaTeX predefined
\DeclareMathOperator{\Lie}{Lie}	% Lie algebra of ...
%
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%
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%
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%
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%	left index(es) in \mathnormal:
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%	Christoffel symbols
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%	Physical Abbreviations:
\newcommand{\ih}{\mathrm{i}\hbar}% imaginary unit times Planck constant/{2\pi}
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%	Sans Serif capital Latin letters & their possible usage
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					% or of the tensor algebra
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					% transport I along paths
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%	Black board capital Latin letters & their possible usage
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%	Latin abbreviations:
\newcommand{\ie} {i.e.\ }		%  That is (Latin: It Est)
\newcommand{\cf} {cf.\ }		%  Compare with (Latin: ConFer)
\newcommand{\eg} {e.g.\ }		%  For example (Latin: Exapli gratia)
\newcommand{\etc}{etc.\ }		%  And so on (Latin: et cetera)
\newcommand{\viz}{viz.\ }		%  That is (Latin: vidilizet)
\newcommand{\vv}{v.v.\ }		%  Inversely (Latin: vice versa)
\newcommand{\vive}{\emph{vice versa}}	%  Inversely (Latin: vice versa)

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%	Misc
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\newcommand{\xlrarrow}[2][12mu]{% extensible left-right arrow over an object
    \overset{%
	\xleftarrow{\hphantom{#2}\mspace{-2mu}}%
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%  >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<


%	>=================================>
%	BEGINNING of Mathematical Operators  (requires amsopn.sty package)

% Set theory and topology
 \DeclareMathOperator{\diam}{diam}	% diameter

% Mappings
 \DeclareMathOperator{\Ker}{Ker}	% Kernel of a mapping
% \DeclareMathOperator{\ker}{ker}	% kernel	; LaTeX predefined
\DeclareMathOperator{\Image}{\mathrm{Im}}% Image of a mapping
 \DeclareMathOperator{\rank}{rank}	% rank
 \DeclareMathOperator{\im}{im}		% image
%\DeclareMathOperator{\Im}{Im}		% Image;LaTeX predefined,Complex numbers
 \DeclareMathOperator{\degree}{deg}	% degree ;-\deg which is LaTeX predefined
 \DeclareMathOperator{\pr}{pr}		% projection
 \DeclareMathOperator{\End}{End}	% set of Endomorphisms
 \DeclareMathOperator{\Aut}{Aut}	% set of Automorphisms

% Complex numbers:
 \DeclareMathOperator{\Imp}{Im}		% Imaginary part; LaTeX predefined
 \DeclareMathOperator{\Rep}{Re}		% Real part     ; LaTeX predefined

% Matrices/Linear operators
 \DeclareMathOperator{\tr}{tr}		% Trace (spur) of matrices/operators
 \DeclareMathOperator{\Tr}{Tr}		% Trace (spur) of matrices/operators
 \DeclareMathOperator{\Spur}{Sp}	% Spur (trace) of matrices/operators
 \DeclareMathOperator{\diag}{diag}	% Diagonal matrix

% Manifolds
 \DeclareMathOperator{\Tan}{T}		% the Tangent functor

% Fibre bundles
 \DeclareMathOperator{\Sec}{Sec}	% set of Sections of a fibre bundle
 \DeclareMathOperator{\PSec}{PSec}	% set of Sections of ... along Paths
 \DeclareMathOperator{\Morf}{Mor}	% set of Morphisms of a fibre bundle
 \DeclareMathOperator{\PMorf}{PMor}	% set of Morphisms of... along Paths
 \DeclareMathOperator{\MorfSec}{MorSec}	% set of Morphism of the set of
					% Sections of a bundle
 \DeclareMathOperator{\Diff}{Diff}	% set of Diffeomorphisms
 \DeclareMathOperator{\Lift}{Lift}	% set of Liftings
 \DeclareMathOperator{\PLift}{PLift}	% set of Liftings along Paths
 \DeclareMathOperator{\Path}{P}		% set of Paths
 \DeclareMathOperator{\PF}{PF}		% set of Functions along Paths
% \DeclareMathOperator{\pr}{\mathrm{pr}}% projection mapping, defined above
 \DeclareMathOperator{\morf}{mor}	% Morphism fibre bundle of a bundle

% Exponents
%\DeclareMathOperator{\exp}{exp}	% exponent: LaTeX predefined
 \DeclareMathOperator{\Exp}{Exp}	% exponent
 \DeclareMathOperator{\Pexp}{Pexp}	% Path/Time ordered exponent
 \DeclareMathOperator{\Texp}{Texp}	% Time/Path ordered exponent

%	Differential vector operators:
\DeclareMathOperator{\diver}{div}	% divergence
\DeclareMathOperator{\grad}{grad}	% gradient
\DeclareMathOperator{\rot}{rot}		% rotor

%	Ordinary and Partial differentials: 	these are defined above
% \DeclareMathOperator{\odo}{\mathrm{d}}  % Ordinary Differential Operator
% \DeclareMathOperator{\pdo}{\partial}    % Partial Differential Operator

% Misilaneous
 \DeclareMathOperator{\sign}{sign}	% sign of a real number
 \DeclareMathOperator{\sgn} {sgn}	% sign of a real number
 \DeclareMathOperator{\Supp}{Supp}	% Support of a function
 \DeclareMathOperator{\supp}{supp}	% Support of a function

%	<==================================<
%	END of Mathematical Operators  (requires amsopn.sty package)



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%***************************************************************************%
%**       END of LaTeX2e style package of BOZHIDAR ZAKHARIEV ILIEV        **%
%***************************************************************************%
    \end{filecontents}
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% Start of the preambule
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      [2006/08/22 Presentation using TeXPower]

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%-----------------------------------------------------------------------------------------------------------------
% Finally, everything is set up. Here we go...
%
\begin{document}

% Some commands
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\begin{slide}
% the \end{slide} is at the end of all slides in the present group of slides
  \title
  {% Main Title
    Normal frames and linear transports\\
	along paths in line bundles.
\\
    {% Subtitle
      \normalfont
      Applications to classical electrodynamics
      }%
    }


\author{
Bozhidar Z.\ Iliev\\
\hspace*{-0.6em}
Laboratory of Mathematical Modeling in Physics\\
\hspace*{-0.7em}
Institute for Nuclear Research and Nuclear Energy\\
Bulgarian Academy of Sciences\\
% Boul.\ Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria
\url{E-mail: bozho@inrne.bas.bg}\\
\href{http://theo.inrne.bas.bg/~bozho/}
{URL: http://theo.inrne.bas.bg/$\sim$bozho/}
}

\date{%
\begin{center}\highlighttext{8$^\text{th}$ International Workshop on\\Complex
			     structures and vector fields
} \end{center}
\vspace{-0.5ex}\normalfont
\highlightenhanced{
\href{http://theo.inrne.bas.bg/csvf2006/}{URL: http://theo.inrne.bas.bg/csvf2006/}
}
}

  \maketitle


%--------------------------------------
\newslide
%--------------------------------------


  \minisec{Abstract}% some title on a slide


    \begin{itemize}
\pageTransitionSplitHO% Split Horizontally Outside
\item
	The definitions and some basic properties of the linear transports
along paths in vector bundles and the normal frames for them are recalled.
 \pause
 \item
	The formalism is specified on line bundles and applied to a
geometrical description of the classical electrodynamics.
\pause
\item
	The inertial frames for this theory are discussed.

    \end{itemize}

%--------------------------------------
\newslide
%--------------------------------------


 % Table of contnets slide(s)
  \tableofcontents

\end{slide}


%%%%%%%%%%%
\begin{slide}
%-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->>
%			BEGINNING OF SECTION~\ref{Sect1}
\section{Linear transports along paths}
\pageTransitionSplitHI% Split Horizontally Inside



	Let $(E,\pi,B)$ be a complex
vector bundle with bundle (total) space $E$, base
$B$, projection $\pi\colon E\to B$, and homeomorphic fibres
$\pi^{-1}(x)$, $x\in B$.
The base $B$ is supposed to be  a $C^1$ differentiable manifold.
By $J$ and $\gamma\colon J\to B$ are denoted real interval  and path in $B$,
respectively. The paths considered are generally \emph{not} supposed to be
continuous or differentiable unless their differentiability class is stated
explicitly. If $\gamma$ is a $C^1$ path, the vector field tangent to it is
denoted by $\dot{\gamma}$.

%--------------------------------------
\newslide
%--------------------------------------


	\begin{Defn}	\label{4-Defn2.1}
	\index{linear transport along paths!definition|defined}
	A \emph{linear transport along paths} in a bundle $(E,\pi,B)$ is a
map $L$ assigning to every path $\gamma$ a map $L^\gamma$,
\emph{transport along} $\gamma$, such that
$L^\gamma\colon (s,t)\mapsto L^\gamma_{s\to t}$ where the map
	\begin{equation}	\label{4-2.1}
L^\gamma_{s\to t} \colon  \pi^{-1}(\gamma(s)) \to \pi^{-1}(\gamma(t))
	\qquad s,t\in J,
	\end{equation}
called \emph{transport along $\gamma$ from $s$ to} $t$, has the properties:
	\begin{alignat}{2}	\label{4-2.2}
L^\gamma_{s\to t}\circ L^\gamma_{r\to s} &=
			L^\gamma_{r\to t},&\  r,s,t&\in J, \\
L^\gamma_{s\to s} &= \id_{\pi^{-1}(\gamma(s))}, & s&\in J,	\label{4-2.3}
\\
L^\gamma_{s\to t}(\lambda u + \mu v) 				\label{4-2.4}
  &= \lambda L^\gamma_{s\to t}u + \mu L^\gamma_{s\to t}v,
	& \lambda,\mu &\in \mathbb{C},
\\\notag && u,v&\in{\pi^{-1}(\gamma(s))},
	\end{alignat}
where  $\circ$ denotes composition of maps and $\id_X$ is the identity map of
a set $X$.
	\end{Defn}

\newpage

	Let $\{e_i(s;\gamma)\}$ be a $C^1$ basis in $\pi^{-1}(\gamma(s))$,
$s\in J$;
here and henceforth the Latin indices run from 1 to
$\dim \pi^{-1}(x),\ x\in B$. We also assume the Einstein summation rule on
indices repeated on different levels.
So, along $\gamma\colon J\to B$ we have a set $\{e_i\}$ of bases on
$\pi^{-1}(\gamma(J))$ such that the liftings $\gamma\mapsto e_i(\cdot,\gamma)$
of paths are of class $C^1$.

	The \emph{matrix}
 $\Mat{L}(t,s;\gamma):=\bigl[\Sprindex[L]{j}{i}(t,s;\gamma)\bigr]$
\emph{(along $\gamma$ at $(s,t)$ in $\{e_i\}$) of a linear transport} $L$
along $\gamma$ from $s$ to $t$ is defined via the expansion
% 	\begin{equation}	\label{4-2.8}
\(
L_{s\to t}^{\gamma} \bigl(e_i(s;\gamma)\bigr)
		=:\Sprindex[L]{i}{j}(t,s;\gamma) e_j(t;\gamma)
		\ s,t\in J.
\)
% 	\end{equation}


%--------------------------------------
\newslide
%--------------------------------------


	\begin{Prop}	\label{4-Prop2.4}
	A \ non\ndash degenerate \ matrix\ndash valued \ function\
$\Mat{L}\colon (t,s;\gamma)\mapsto\Mat{L}(t,s;\gamma)$
is a matrix of some linear transport along paths $L$ (in a given field
$\{e_i\}$ of bases along $\gamma$) iff
	\begin{equation}	\label{4-2.14}
\Mat{L}(t,s;\gamma) = \Mat{F}^{-1}(t;\gamma) \Mat{F}(s;\gamma)
	\end{equation}
where $\Mat{F}\colon (t;\gamma)\mapsto\Mat{F}(t;\gamma)$ is a non\ndash degenerate
matrix\nobreakdash-valued function.
	\end{Prop}

\newpage

	\begin{Prop}	\label{4-Prop2.5}
	If the matrix $\Mat{L}$ of a linear transport $L$ along paths has a
some representation
% 	\begin{equation}	\label{4-2.15}
\(
\Mat{L}(t,s;\gamma)
   = \lindex[\mspace{-2mu}\Mat{F}]{}{\star}^{-1}(t;\gamma)
     \lindex[\mspace{-2mu}\Mat{F}]{}{\star}(s;\gamma)
\)
% 	\end{equation}
for some matrix-valued function
$\lindex[\mspace{-2mu}\Mat{F}]{}{\star}(s;\gamma)$,
then all matrix\nobreakdash-valued functions $\Mat{F}$ representing $\Mat{L}$
via~\eref{4-2.14} \ are \ given \ by
 	\begin{equation}	\label{4-2.16}
% \(
\Mat{F}(s;\gamma) = \Mat{D}^{-1}(\gamma)
   \lindex[\mspace{-2mu}\Mat{F}]{}{\star}(s;\gamma)
% \)
 	\end{equation}
where $\Mat{D}(\gamma)$ is a non\ndash degenerate matrix depending only on $\gamma$.
	\end{Prop}

%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionSplitVO% Split Vertically Outside


	Let $\{e_i(s;\gamma)\}$ be a smooth field of bases along
$\gamma\colon J\to B,\ s\in J$. The explicit local action of the derivation
$D\colon\gamma\mapsto D^\gamma \colon s\mapsto D_s^\gamma$, associated to $L$,
on a $C^1$ lifting of paths $\lambda$ is
	\begin{equation}	\label{4-2.22}
D_s^\gamma\lambda
   = \biggl[ \frac{\od \lambda_\gamma^i(s)}{\od s}
     + \Sprindex[\Gamma]{j}{i}(s;\gamma)\lambda_\gamma^j(s)
     \biggr]  e_i(s;\gamma) .
	\end{equation}
\pause
Here the \emph{(2-index) coefficients}
$\Sprindex[\Gamma]{j}{i}$ of the linear transport $L$ are defined by
	\begin{equation}	\label{4-2.23}
\Sprindex[\Gamma]{j}{i}(s;\gamma)
   := \frac{\pd\Sprindex[L]{j}{i}(s,t;\gamma)}{\pd t}\bigg|_{t=s}
   = - \frac{\pd\Sprindex[L]{j}{i}(s,t;\gamma)}{\pd s}\bigg|_{t=s}
	\end{equation}
and, evidently, uniquely determine the derivation $D$ generated by $L$.

%--------------------------------------
\newslide
%--------------------------------------


If a matrix $\Mat{F}$ determines the matrix $\Mat{L}$ of a transport $L$
according to proposition~\ref{4-Prop2.4}, then
	\begin{equation}	\label{4-2.25}
\Mat{\Gamma}(s;\gamma):=\bigl[ \Sprindex[\Gamma]{j}{i}(s;\gamma) \bigr]
   = \genfrac{.}{|}{}{}{\pd\Mat{L}(s,t;\gamma)}{\pd t}_{t=s}
   = \Mat{F}^{-1}(s;\gamma)\frac{\od \Mat{F}(s;\gamma)}{\od s} .
	\end{equation}

\pause
A change
$\{e_i\}\to\{e_{i}^{\prime}=A_{i}^{j}e_i\}$ of the bases along a path $\gamma$
with a non\ndash degenerate $C^1$ matrix\ndash valued function
$A(s;\gamma):=\bigl[A_{i}^{j}(s;\gamma)\bigr]$ implies
\(
\Mat{\Gamma}(s;\gamma) = \bigl[ \Sprindex[\Gamma]{j}{i}(s;\gamma) \bigr]
	\mapsto  \Mat{\Gamma}^\prime(s;\gamma)
   = \bigl[ \Sprindex[\Gamma]{j}{\prime\,i}(s;\gamma) \bigr]
\)
with
	\begin{equation}	\label{4-2.26}
\Mat{\Gamma}^\prime(s;\gamma)
  = A^{-1}(s;\gamma) \Mat{\Gamma}(s;\gamma) A(s;\gamma)
    + A^{-1}(s;\gamma)\frac{\od A(s;\gamma)}{\od s}.
	\end{equation}

Reconstruction of $L$ from $\Mat{\gamma}$ ...


%			END OF SECTION~\ref{Sect1}
%<<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<
\end{slide}


%-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->>
\begin{slide}
 \pageTransitionSplitVI% Split Vertically Inside

%			BEGINNING OF SECTION~\ref{Sect2}
\section{Normal frames for linear transports}


	Let a linear transport $L$ along paths be given in a vector bundle
$(E,\pi,B)$, $U\subseteq B$ be an arbitrary subset in $B$, and
$\gamma\colon J\to U$ be a path in $U$.

%--------------------------------------
\newslide
%--------------------------------------

	\begin{Defn}	\label{4-Defn3.1}
\index{normal frame!in vector bundles!basic definitions|(}
	A \emph{frame field (of bases)} in $\pi^{-1}(\gamma(J))$ is called
\emph{normal along $\gamma$ for} $L$ if the matrix of $L$ in it is the
identity matrix along the given path $\gamma$.
% 	\end{Defn}
%
% 	\begin{Defn}	\label{4-Defn3.2}
	A \emph{frame field (of bases)} defined on $U$ is
called \emph{normal on $U$ for} $L$ if it is normal along every path
$\gamma\colon J\to U$ in $U$. The \emph{frame} is called \emph{normal for}
$L$ if $U=B$.
\index{normal frame!in vector bundles!basic definitions|)}
	\end{Defn}

\pause

	\begin{Defn}	\label{4-Defn3.3}
\index{linear transport along paths!Euclidean!basic definitions|(}
	A \emph{linear transport along paths} (or \emph{along a path $\gamma$})
is called \emph{Euclidean along some (or the given) path} $\gamma$ if it admits
a frame normal along $\gamma$.
% 	\end{Defn}
%
% 	\begin{Defn}	\label{4-Defn3.4}
	A \emph{linear transport along paths} is called \emph{Euclidean} on
$U$ if it admits frame(s) normal on $U$. It is called \emph{Euclidean} if
$U=B$.
\index{linear transport along paths!Euclidean!basic definitions|)}
	\end{Defn}


%--------------------------------------
\newslide
%--------------------------------------


	\begin{Prop}	\label{4-Prop3.1}
	The following statements are equivalent in a given frame $\{e_i\}$
over $U\subseteq B$:

\begin{subequations}	\label{4-3.1}
\indent
\textup{\textbf{\hphantom{v}(i)}}
The matrix of $L$ is the identity matrix on $U$, \ie
% 	\begin{equation}	\label{4-3.1a}
\(
\Mat{L}(t,s;\gamma) = \bs{1}%\openone
\)
% 	\end{equation}
along every path $\gamma$ in $U$.

\textup{\textbf{\hphantom{i}(ii)}}
The matrix of $L$ along every $\gamma\colon J\to U$
depends only on $\gamma$, \ie it is independent of the points at which it is
calculated:
% 	\begin{equation}	\label{4-3.1b}
\(\Mat{L}(t,s;\gamma) = C(\gamma)
\)
% 	\end{equation}
where $C$ is a matrix-valued function of $\gamma$.

\textup{\textbf{\hphantom{}(iii)}}
If $E$ is a $C^1$ manifold, the coefficients
$\Sprindex[\Gamma]{j}{i}(s;\gamma)$ of $L$ vanish on $U$, \ie
% 	\begin{equation}	\label{4-3.1c}
\(\Mat{\Gamma}(s;\gamma) = 0
\)
% 	\end{equation}
along every path $\gamma$ in $U$.

\textup{\textbf{\hphantom{}(iv)}}
The explicit local action of the derivation $D$ along paths generated by $L$
reduces on $U$ to differentiation of the components of the liftings with
respect to the path's parameter if the path lies entirely in $U$:
% 	\begin{equation}	\label{4-3.1d}
\(
D_{s}^{\gamma}\lambda
		= \frac{\od\lambda_\gamma^i(s)}{\od s} \, e_i(s;\gamma)
\)
% 	\end{equation}
where $\lambda=\lambda^ie_i$
is a $C^1$ lifting of paths and
$\lambda\colon\gamma\mapsto\lambda_\gamma$.

\textup{\textbf{\hphantom{i}(v)}}
The transport $L$ leaves the vectors' components
unchanged along any path in $U$, \viz we have
% 	\begin{equation}	\label{4-3.1e}
\(
L_{s\to t}^{\gamma}\bigl(u^i e_i(s;\gamma)\bigr) = u^i e_i(t;\gamma)
\)
% 	\end{equation}
for all $u^i\in\mathbb{C}$.

\textup{\textbf{\hphantom{}(vi)}}
The basic vector fields are $L$\ndash transported
along any path $\gamma\colon J\to U$:
% 	\begin{equation}	\label{4-3.1f}
\(
L_{s\to t}^{\gamma} \bigl(e_i(s;\gamma)\bigr) =e_i(t;\gamma).
\)
%	\end{equation}
	\end{subequations}
	\end{Prop}


%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionBlindsH% Horizontal Blinds


	\begin{Cor}	\label{4-Cor4.1}
	Every linear transport along paths is Euclidean along every fixed
path without self\ndash intersections.
	\end{Cor}

\pause

	\begin{Thm}	\label{4-Thm3.1}
	A linear transport along paths admits frames normal on some set
(resp.\ along a given path) if and only if its action along every path in
this set (resp.\ along the given path) depends only on the initial and final
point of the transportation but not on the particular path connecting these
points. In other words, a transport is Euclidean on $U\subseteq B$ iff it
is path\ndash independent on $U$.
	\end{Thm}


%--------------------------------------
\newslide
%--------------------------------------


	\begin{Prop}	\label{4-Prop5.1}
Let $L$ be a transport along paths in a bundle $(E,\pi,M)$, $E$ and $M$
being $C^1$ manifolds, and $L$  be Euclidean on $U\subseteq M$ (resp.\ along a
$C^1$ path $\gamma\colon J\to M$). Then the matrix $\Mat{\Gamma}$ is of the
form
% of its coefficients has the representation
	\begin{equation}	\label{4-5.1}
\Mat{\Gamma}(s;\gamma)
   = \sum_{\mu=1}^{\dim M} \Gamma_\mu(\gamma(s)) \dot\gamma^\mu(s)
   \equiv \Gamma_\mu(\gamma(s)) \dot\gamma^\mu(s)
	\end{equation}
in any frame $\{e_i\}$ along every (resp.\ the given) $C^1$ path
$\gamma\colon J\to U$, where
\(
{\Gamma}_\mu =
  \bigl[\Sprindex[\Gamma]{j\mu}{i}\bigr]_{i,j=1}^{\dim\pi^{-1}(x)}
\)
are some matrix-valued functions, defined on an open set~$V$ containing $U$
(resp.\ $\gamma(J)$) or equal to it, and $\dot\gamma^\mu$ are the components
of $\dot\gamma$ in some frame $\{E_\mu\}$ along $\gamma$ in the bundle space
tangent to $M$, $\dot\gamma=\dot\gamma^\mu E_\mu$. The functions
$\Sprindex[\Gamma]{j\mu}{i}$ are termed \emph{3\ndash index coefficients} of
$L$.
	\end{Prop}

%--------------------------------------
\newslide
%--------------------------------------


	Let $U$ be an open set, \eg $U=M$. If we change the frame $\{E_\mu\}$
in the bundle space tangent to $M$,
$\{E_\mu\}\mapsto \{E_\mu^{\prime} = B_{\mu}^{\nu}E_\nu\}$ with
$B=\bigl[B_{\mu}^{\nu}\bigr]$ being non\ndash degenerate matrix\ndash valued
function, and simultaneously the bases in the fibres $\pi^{-1}(x)$, $x\in M$,
$\{e_i|_x\} \mapsto \{e_i^{\prime}|_x = A_{i}^{j}(x)e_j|_x\}$, then,
from~\eref{4-2.26} and~\eref{4-5.1}, we see that ${\Gamma}_\mu$ transforms
into ${\Gamma}_\mu^{\prime}$ such that
	\begin{equation}	\label{4-5.3}
{\Gamma}_\mu^{\prime}
  = B_{\mu}^{\nu} A^{-1}{\Gamma}_\nu A  + A^{-1} E'_\mu(A)
  = B_{\mu}^{\nu} A^{-1}\bigl( {\Gamma}_\nu A  + E_\nu(A) \bigr)
	\end{equation}
where $A:=\bigl[A_{i}^{j}\bigr]_{i,j=1}^{\dim\pi^{-1}(x)}$ is
non\ndash degenerate and of class $C^1$.

%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionBlindsV% Vertical Blinds



	\begin{Thm}	\label{4-Thm5.1}
	A $C^2$ linear transport $L$ along paths is Euclidean on a
neighborhood $U\subseteq M$ if and only if in every frame the matrix
$\Mat{\Gamma}$ of its coefficients has a representation~\eref{4-5.1} along
every $C^1$ path $\gamma$ in $U$ in which the matrix\nobreakdash-valued
functions $\Gamma_\mu$, defined on an open set containing $U$ or equal to it,
satisfy the equalities
	\begin{equation}	\label{4-5.5}
\bigl( R_{\mu\nu}(-{\Gamma}_1,\ldots,-{\Gamma}_{\dim M}) \bigr) (x)
  = 0
	\end{equation}
where $x\in U$ and
	\begin{equation}	\label{4-5.6}
R_{\mu\nu}(-{\Gamma}_1,\ldots,-{\Gamma}_{\dim M})
  := - \frac{\pd{\Gamma}_\mu}{\pd x^\nu}
     + \frac{\pd{\Gamma}_\nu}{\pd x^\mu}
     + {\Gamma}_\mu{\Gamma}_\nu
     - {\Gamma}_\nu{\Gamma}_\mu .
\end{equation}
in a coordinate frame
$\bigl\{E_\mu=\frac{\pd}{\pd x^\mu}\bigr\}$ in a neighborhood of $x$.
	\end{Thm}

%--------------------------------------
\newslide
%--------------------------------------


	\begin{Thm}	\label{4-Thm5.2}
A linear transport $L$ along paths is Euclidean on a submanifold $N$ of $M$
if and only if in every frame $\{e_i\}$, in the bundle space over $N$, the
matrix of its coefficients has a representation~\eref{4-5.1} along every
$C^1$ path in $N$ and, for every $p_0\in N$ and a chart $(V,x)$ of  $M$
such that $V\ni p_0$ and
$x(p)=( x^1(p),\dots,x^{\dim N}(p),t_0^{\dim N+1},\dots,t_0^{\dim M} )$
for every $p\in N\cap V$ and constant numbers
$t_0^{\dim N+1},\dots,t_0^{\dim M}$, the equalities
	\begin{equation}	\label{4-5.14}
\bigl(  R^N_{\alpha\beta}(-\Gamma_1,\ldots,-\Gamma_{\dim N}) \bigr) (p) = 0,
\qquad
\alpha,\beta=1,\dots,\dim N
	\end{equation}
hold for all $p\in N\cap V$ and
	\begin{multline}	\label{4-5.15}
%	\begin{equation}	\label{4-5.15}
R^N_{\alpha\beta}(-\Gamma_1,\ldots,-\Gamma_{\dim N})
%\\
:= R_{\alpha\beta}(-\Gamma_1,\ldots,-\Gamma_{\dim M})
\\
 =   - \frac{\pd \Gamma_\alpha} {\pd x^\beta}
     - \frac{\pd \Gamma_\beta}  {\pd x^\alpha}
     + \Gamma_\alpha \Gamma_\beta
     - \Gamma_\beta  \Gamma_\alpha .
%	\end{equation}
	\end{multline}
Here $\Gamma_1,\ldots,\Gamma_{\dim N}$ are the first $\dim N$ of the matrices of
the 3-index coefficients of $L$ in the coordinate frame
$\bigl\{\frac{\pd}{\pd x^\mu}\bigr\}$ in the tangent bundle space over
$N\cap V$.
	\end{Thm}

\end{slide}
%			END OF SECTION~\ref{Sect2}
%<<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<




%-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->>
%			BEGINNING OF SECTION~\ref{Sect3}
\begin{slide}
\pageTransitionBoxO% Growing Box

\section
{Transports and frames in line bundles}


	Let $(E,\pi,M)$ be one-dimensional vector bundle over a $C^1$
manifold $M$; such bundles are called line bundles. Thus the (typical) fibre of
$(E,\pi,M)$ can be identified with $\mathbb{C}$ (resp.\ $\mathbb{R}$ in the
real case) and then the fibre $\pi^{-1}(x)$ over $x\in M$ will be an isomorphic
image of $\mathbb{C}$ (resp.\ $\mathbb{R}$ in the real case). Let $\gamma\colon
J\to M$ be of class $C^1$ and $L$ be a linear transport along paths in
$(E,\pi,M)$. A frame $\{e\}$ along $\gamma$ consists of a single non\ndash zero
vector field $e\colon (s;\gamma)\to
e(s;\gamma)\in\pi^{-1}(\gamma(s))\backslash\{0\}$, $s\in J$, and in it the
matrix of $L^\gamma$ at $(t,s)\in J\times J$ is simply a number
$\Mat{L}(t,s;\gamma)\in\mathbb{C}$,
 $L_{s\to t}^{\gamma}(u e(s;\gamma)) = u \Mat{L}(t,s;\gamma) e(t;\gamma)$
for $u\in\mathbb{C}$ and $s,t\in J$.


%--------------------------------------
\newslide
%--------------------------------------

Te
general form of $\Mat{L}$ is
(proposition~\ref{4-Prop2.4})
	\begin{equation}	\label{4-5n.1}
\Mat{L}(t,s;\gamma) = \frac{f(s;\gamma)}{f(t;\gamma)}
	\end{equation}
where $f\colon (s;\gamma)\mapsto f(s;\gamma)\in\mathbb{C}\backslash\{0\}$ is
defined up to (left) multiplication with a function of $\gamma$
(proposition~\ref{4-Prop2.5}). Respectively, due to~\eref{4-2.25}, the matrix
of the coefficient(s) of $L$ is
	\begin{equation}	\label{4-5n.2}
\Mat{\Gamma}(s;\gamma)
= \frac{\pd \Mat{L}(t,s;\gamma)}{\pd s}\Big|_{t=s}
= \frac{1}{f(s;\gamma)} \frac{\od f(s;\gamma)}{\od s}
= \frac{\od}{\od s}\bigl[ \ln(f(s;\gamma)\bigr]
	\end{equation}
and~ the transport's matrix takes the form
	\begin{equation}	\label{4-5n.3}
\Mat{L}(t,s;\gamma)
= \exp \biggl(
	- \int\limits_{s}^{t} \Mat{\Gamma}(\sigma;\gamma) \Id\sigma
	\biggr).
	\end{equation}


%--------------------------------------
\newslide
%--------------------------------------

	A change $e(s;\gamma)\mapsto e'(s;\gamma)=a(s;\gamma)e(s;\gamma)$,
with $a(s;\gamma)\in\mathbb{C}\backslash\{0\}$, of the frame $\{e\}$ implies
% (see~\eref{4-2.10} and~\eref{4-2.26})
	\begin{subequations}	\label{4-5n.4}
	\begin{gather}	\label{4-5n.4a}
\Mat{L}(t,s;\gamma) \mapsto \Mat{L}'(t,s;\gamma)
	= \frac{a(s;\gamma)}{a(t;\gamma)} \Mat{L}(t,s;\gamma)
\\	\label{4-5n.4b}
\Mat{\Gamma}(s;\gamma) \mapsto \Mat{\Gamma}'(s;\gamma)
	= \Mat{\Gamma}(s;\gamma) +
		\frac{\od}{\od s}\bigl[ \ln(a(s;\gamma)\bigr] .
	\end{gather}
	\end{subequations}

	The explicit local action of the derivation $D$ along paths generated
by $L$ is
	\begin{equation}	\label{4-5n.5}
D_{s}^{\gamma} \lambda
= \Bigl(
	\frac{\od\lambda_\gamma(s)}{\od s}
	+ \Mat{\Gamma}(s;\gamma) \lambda_\gamma(s)
  \Bigr) e(s;\gamma)
	\end{equation}
where $\lambda\in\PLift^1(E,\pi,M)$ and~\eref{4-2.22} was used.

%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionBoxI% Shrinking Box


	\minisec{Normal frames on one-dimensional vector bundles}


%--------------------------------------
%\newslide
%--------------------------------------



	A frame $\{e\}$ is normal for $L$ along $\gamma$ (resp.\ on $U$) iff
in that frame equation~\eref{4-5n.1} holds with
	\begin{equation}	\label{4-5n.6}
f(s;\gamma) = f_0(\gamma)
	\end{equation}
where  $\gamma\colon J\to M$ (resp.\ $\gamma\colon J\to U$) and
$f_0\colon\gamma\mapsto f_0(\gamma)\in\mathbb{C}\backslash\{0\}$
Since, in a frame
normal along $\gamma$ (resp.\ on $U$), it is fulfilled
	\begin{equation}	\label{4-5n.7}
\Mat{L}(t,s;\gamma) = \openone,
\quad
\Mat{\Gamma}(s;\gamma) = 0
	\end{equation}
for the given path $\gamma$ (resp.\ every path in $U$),
in every frame $\{e'=ae\}$, we have
	\begin{equation}	\label{4-5n.8}
\Mat{L}'(t,s;\gamma) = \frac{a(s;\gamma)}{a(t;\gamma)},
\quad
\Mat{\Gamma}'(s;\gamma) = \frac{\od}{\od s}\bigl[ \ln(a(s;\gamma)\bigr] .
	\end{equation}

%--------------------------------------
\newslide
%--------------------------------------

In addition, for Euclidean on $U\subseteq M$ transport $L$, the representation
	\begin{equation}	\label{4-5n.9}
\Mat{\Gamma}'(s;\gamma) = \Gamma'_\mu(\gamma(s)) \dot\gamma^{\prime\,\mu}(s)
	\end{equation}
holds for every $C^1$ path $\gamma\colon J\to U$ and some
$\Gamma'_\mu\colon V\to\mathbb{C}$ with $V$ being an open set such that
$V\supseteq U$ (proposition~\ref{4-Prop5.1}). This means (see
theorem~\ref{4-Thm3.1}) % and~\cite[theorem~\ref{LB-4-Thm3.2}]{bp-NF-LTP})
that~\eref{4-5n.8} holds for
	\begin{equation}	\label{4-5n.10}
a(s;\gamma)=a_0(\gamma(s)),
	\end{equation}
where $a_0\colon U\to\mathbb{C}\backslash\{0\}$, and, consequently, the
equality~\eref{4-5n.9} can be satisfied if we choose
	\begin{equation}	\label{4-5n.11}
\Gamma'_\mu = E_\mu(a)
	\end{equation}
with $a\colon V\to\mathbb{C}$, $a|_U=a_0$ and $\{E_\mu\}$ being a frame in
the bundle space tangent to $M$ which, in particular, can be a coordinate
one, $E_\mu=\frac{\pd}{\pd x^\mu}$.  Of course, if $U$ is not an open set,
this choice of $\Gamma'_\mu$ is not necessary; for
example, the equality~\eref{4-5n.9} will be preserved, if to the r.h.s.\
of~\eref{4-5n.11} is added a function $G'_\mu$ such that
$G'_\mu\dot\gamma^{\prime\,\mu}=0$.

% %--------------------------------------
% \newslide
% %--------------------------------------
%
%
% 	By virtue of~\eref{4-5.3}, the functions $\Gamma_\mu$ and
% $\Gamma'_\mu$ in two arbitrary pairs of frames
% $(\{e\},\{E_\mu\})$ and
% $(\{e'=ae\},\{E'_\mu=B_\mu^\nu E_\nu\})$, respectively, are connected via
% 	\begin{equation}	\label{4-5n.12}
% \Gamma'_\mu
% = B_\mu^\nu \Gamma_\nu + \frac{1}{a} E'_\mu(a)
% = B_\mu^\nu \bigl( \Gamma_\nu +  E_\nu(\ln a) \bigr)
% 	\end{equation}
% and, consequently, with respect to changes of the frames in the tangent
% bundle space over $M$, when $a=1$, they behave like the components of a
% covariant vector field (one\ndash form).
% Therefore, on an open set $U$, \eg $U=M$, the quantity
% 	\begin{equation}	\label{4-5n.13}
% \omega = \Gamma_\mu E^\mu,
% 	\end{equation}
% where $\{E^\mu\}$ is the coframe dual to $\{E_\mu\}$ (in local coordinates:
% $E_\mu=\frac{\pd}{\pd s^\mu}$ and $E^\mu= \od x^\mu$), is a 1\ndash form
% over $M$ (with respect to changes of the local coordinates on $M$ or of
% the frames in the (co)tangent bundle space over $M$). However, it depends
% on the choice of the frame $\{e\}$ in the bundle space $E$ and a change
% $e\mapsto e'=ae$ implies
% 	\begin{equation}	\label{4-5n.14}
% \omega\mapsto\omega'
% = \omega + (E_\nu(\ln a))E^\nu
% = \omega + (E'_\nu(\ln a))E^{\prime\,\nu} .
% 	\end{equation}
%
% 	Using the 1-form~\eref{4-5n.13}, we see that
% 	\begin{equation}	\label{4-5n.15}
% \Mat{\Gamma}(s;\gamma)=\omega|_{\gamma(s)}(\dot\gamma(s))
% 	\end{equation}
% and~\eref{4-5n.3} can be rewritten as
% 	\begin{equation}	\label{4-5n.16}
% \Mat{L}(t,s;\gamma)
% = \exp \biggl(
% 	- \int\limits_{\gamma(s)}^{\gamma(t)} \omega
% 	\biggr)
% 	\end{equation}
% where the integration is along some path in $U$ (on which the transport $L$ is
% Euclidean). Hence $\Mat{L}$ (or $L$) depends only on the points $\gamma(s)$
% and $\gamma(t)$, not on the particular path connecting them, as it should be
% (theorem~\ref{4-Thm3.1}). The self\ndash consistency of our results is
% confirmed by the equation
% 	\begin{equation}	\label{4-5n.17}
% R_{\mu\nu}|_U = 0
% 	\end{equation}
% which is a consequence of~\eref{4-5n.11} and~\eref{4-5.6} and which is a
% necessary and sufficient condition for the existence of frames normal on an
% open set $U$ (theorem~\ref{4-Thm5.1}).
%

%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionWipe{90}


	Final remark:
    \begin{itemize}
\item\pause{
 Frames normal along injective
paths always exist (corollary~\ref{4-Cor4.1}).

\item
On an arbitrary submanifold
$N\subseteq M$ normal frames exist iff the functions $\Gamma_\mu$ satisfy the
conditions~\eref{4-5.14} with $x\in N$ in the coordinates described in
theorem~\ref{4-Thm5.2}.
}
    \end{itemize}

\end{slide}
%			END OF SECTION~\ref{Sect3}
%<<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<



%-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->>
%			BEGINNING OF SECTION~\ref{Sect4}
\begin{slide}
\pageTransitionWipe{180}
\section
{The classical electromagnetic field}


	Now we would like to apply the above formalism to a description of
the classical electromagnetic field. Before going on, we should say that the
accepted natural formalism in gauge field theories, in particular in the
electrodynamics, is via connections on vector
bundles~\cite{Konopleva&Popov,Drechsler&Mayer}. Below we sketch an equivalent
technique for an electromagnetic field.

%--------------------------------------
\newslide
%--------------------------------------


	Recall, the classical electromagnetic
field is described via a real 1\ndash form $A$ over a 4\ndash dimensional
real manifold $M$ (endowed with a (pseudo-)Riemannian metric $g$ and)
representing the space\ndash time model and, usually, identified with the
Minkowski space $M^4$ of special relativity or the (pseudo\ndash)Riemannian
space $V_4$ of general relativity.
The electromagnetic field itself is represented by the two\ndash form
$F=\od A$, where ``$\od$'' denotes the exterior derivative operator, with
local components (in some local coordinates $\{x^\mu\}$)
	\begin{equation}	\label{4-5n.18}
F_{\mu\nu}
= - \frac{\pd A_\mu}{\pd x^\nu} +  \frac{\pd A_\nu}{\pd x^\mu} .
	\end{equation}

%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionWipe{270}


	As it is well known, the electromagnetic field, the
Maxwell equations describing it,
 and its (minimal) interactions with other objects are
invariant under a gauge transformation
	\begin{equation}	\label{4-5n.19}
A_\mu \mapsto A'_\mu
	= A_\mu + \frac{\pd \lambda}{\pd x^\mu}
	\end{equation}
or $A\mapsto A'=A+\od \lambda$, where $\lambda$ is a $C^2$ function. As is
almost evident, the electromagnetic field is invariant under simultaneous
changes of the local coordinate frame,
 $E_\mu=\frac{\pd}{\pd x^\mu}\mapsto E'_\mu=B_\mu^\nu E_\nu$
with $B_\mu^\nu:=\frac{\pd x^\nu}{\pd x^{\prime\,\mu}}$,
and a gauge transformation~\eref{4-5n.19}:
	\begin{equation}	\label{4-5n.20}
A_\mu \mapsto A'_\mu
	= B_\mu^\nu A_\nu + E'_\mu(\lambda)
	= B_\mu^\nu \Bigl(A_\nu + \frac{\pd \lambda}{\pd x^\nu} \Bigr) .
	\end{equation}

%--------------------------------------
\newslide
%--------------------------------------

Under the transformation~\eref{4-5n.20}, the
quantities~\eref{4-5n.18} transform like components of an antisymmetric tensor,
	\begin{equation}	\label{4-5n.21}
F_{\mu\nu}\mapsto F'_{\mu\nu} = B_\mu^\sigma B_\nu^\tau F_{\sigma\tau}
	\end{equation}
due to which the 2-form $F$ remains unchanged, $F=\od A=\od A'$. Notice,
above $A'_\mu$ are \emph{not} the components of $A$ in $\{E'_\mu\}$ unless
$\lambda=\const$ while  $F_{\mu\nu}^{\prime}$ \emph{are} the components of
$F$ in $\big\{E^{\prime\mu}=\frac{\pd x^{\prime\mu}}{\pd x^\nu} \od
x^{\nu}\big\}$.

% 	The similarity between~\eref{4-5n.20} and~\eref{4-5n.12}
% % (and also between~\eref{4-5n.21} and~\eref{4-5.6new})
% is obvious and implies the idea of
Now there arises an idea for
identifying the  electromagnetic potentials
$A_\mu$ with the matrices $\Gamma_\mu$ of
the 3\ndash index coefficients of some linear transport along paths in a
1\ndash dimensional vector bundle $(E,\pi,M)$. This can be done as follows.


%--------------------------------------
\newslide
%--------------------------------------

	Let $M$ be a real 4-dimensional manifold, representing the
space\ndash time  model, and $(E,\pi,M)$  be a 1\ndash dimensional real
vector bundle over it.
We
\emph{%
identify the potentials $A_\mu$ of an electromagnetic field with the (local)
coefficients of a linear transport $L$ along paths in $(E,\pi,M)$ whose
matrix has the representation %~\eref{4-5n.9}
%	\begin{equation}%	\label{4-5n.9}
\(
\Mat{\Gamma}'(s;\gamma) = \Gamma'_\mu(\gamma(s)) \dot\gamma^{\prime\,\mu}(s)
\)
%	\end{equation}
(along every path and in every
pair of frames)%
}.
Hence, the 3-index coefficients of $L$ are uniquely defined and supposed
to be (arbitrarily) fixed in some pair of frames.


%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionGlitter{0}

% 	Since the 3-index coefficients of linear transport are defined in a
% pair of frames $(\{e\},\{E_\mu\})$, $\{e\}$ in the bundle space $E$ and
% $\{E_\mu\}$ in the tangent bundle space $T(M)$, the change~\eref{4-5n.20}
% expresses simply the transformation of $A_\mu$ under the pair of changes
% $e\mapsto e'=ae$ and $E_\mu\mapsto E'_\mu=B_\mu^\nu E_\nu$  and is a
% consequence of~\eref{4-5n.12} if we put
% 	\begin{equation}	\label{4-5n.22}
% a=\e^\lambda .
% 	\end{equation}
It should be emphasized, now the (pure) gauge transformation~\eref{4-5n.19}
appears as a special case of~\eref{4-5n.20}, corresponding to a change of
the frame in $E$ and a fixed frame in $T(M)$.
This means that, in the approach proposed, the %change~\eref{4-5n.19} is
gauge transformations are
directly incorporated in the definition of the field potential $A$. This
conclusion is in contrast to the situation in classical electrodynamics as
in it the gauge transformations are a simple observation of `additional'
invariance of the field, which is not connected with the geometrical
interpretation of the theory.

%--------------------------------------
\newslide
%--------------------------------------


	Defining the electromagnetic field by $F=\od A$,
the equality~\eref{4-5n.18} remains valid in a coordinate frame
$\{E_\mu=\pd/\pd x^\mu\}$. Since $A$ and $F$ possess all of the properties
they must have in electrodynamics, they represent an equivalent
description of electromagnetic field. The only difference with respect to
the classical description is the clear geometrical meaning of these
quantities, as a consequence of which an electromagnetic field can be
identified with a linear transport along paths in a one\ndash dimensional
vector bundle over the space\ndash time.
	The proposed treatment of
electromagnetic field is equivalent to the modern one in the bundle picture
of gauge theories~\cite{Konopleva&Popov,Baez&Muniain},
where the electromagnetic potentials are regarded as coefficients of a
suitable linear connection.


%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionGlitter{315}

	In the approach proposed, the different gauge conditions, which are
frequently used, find a natural interpretation as a
\highlighttext{partial fix of the class of frames in the bundle space}
 employed.\\
%
For instance, any one of the gauges
in the table~\ref{4-TableOfGauges} on the next slide corresponds to a
class of frames for which~\eref{4-5n.20} holds for $B_\mu^\nu=\delta_\mu^\nu$,
$\delta_\mu^\nu$ being the Kronecker deltas, and $\lambda$ subjected to a
condition given in the table.
(%
Below $M$ is supposed  to be endowed with a (pseudo-)Riemannian metric $g_{\mu\nu}$,
the coordinates to be numbered as $x^0,x^1,x^2$, and $x^3$, $x^0$ to be the
`time' coordinate, $\pd_\mu:=\pd/\pd x^\mu$, and $\pd^\mu:=g^{\mu\nu}\pd_\nu$
with $[g^{\mu\nu}]:=[g_{\mu\nu}]^{-1}$.%
)

\vspace{-2.3ex}
	\begin{table}[ht!]
\caption{Examples of gauge conditions \label{4-TableOfGauges}}
\vspace{1ex}
	\begin{minipage}{\textwidth}
	\begin{tabularx}{\textwidth}{lrl@{}}%{lrll@{}}
%	\begin{tabular}{lrll}
Gauge  & Condition on $A$  & Condition on $\lambda$% & Condition on $\varphi$
\\ \hline
Lorenz~%
% \footnote{~%\normalfont\footnotesize%
% The Lorenz condition and gauge
% are named in honor of the Danish theoretical physicist
% Ludwig Valentin Lorenz (1829--1891),
% who has first published it in 1867~\cite{Lorenz/1867}
% (see also~\cite[pp.\ 268-269, 291]{Whittaker-History});
% however this condition was first introduced in lectures by Bernhard
% G.~W.~Riemann in 1861 as pointed in~\cite[p.~291]{Whittaker-History}. It
% should be noted that the \emph{Lorenz} condition/gauge is quite often
% erroneously referred to as the Loren\emph{t}z condition/gauge after the name of
% the Dutch theoretical physicist Hendrik Antoon Lorentz (1853--1928) as, e.g.,
% in~\cite[p.~18]{Roman-QFT} and in~\cite[p.~45]{Gockeler&Schucker}.%
% }
	& $\pd^\mu A_\mu =0$
	& $\pd^\mu\pd_\mu \lambda =0$
% 	& $\pd^\mu\pd_\mu \varphi = - \pd^\mu\pd_\mu \lambda $
\\
Coulomb\footnote{In this row the summation over $k$ is from 1 to 3.}
%	& $\sum_{k=1}^{3}\pd^k A_k =0$
%	& $\sum_{k=1}^{3}\pd^k \pd_k \lambda =0$
%	& $\sum_{k=1}^{3}\pd^k \pd_k \varphi
%		=- \sum_{k=1}^{3}\pd^k \pd_k \lambda $
	& $\pd^k A_k =0$
	& $\pd^k \pd_k \lambda =0$
% 	& $\pd^k \pd_k \varphi
% 		=- \pd^k \pd_k \lambda $
\\
Hamilton
	& $A_0=0$
	& $\lambda(x) = \lambda(x^1,x^2,x^3)$
% 	& $\varphi(x) = \varphi(x^1,x^2,x^3)$
\\
Axial
	& $A_3=0$
	& $\lambda(x) = \lambda(x^0,x^1,x^2)$
% 	& $\varphi(x) = \varphi(x^0,x^1,x^2)$
\\ \hline
%	\end{tabular}
	\end{tabularx}
	\end{minipage}
 	\end{table}
% In the table~\ref{4-TableOfGauges} $\varphi$ is a
% $C^1$ function describing the arbitrariness in the choice of $\lambda$, \ie if
% a gauge condition is valid for $\lambda$, then it holds also for
% $\lambda+\varphi$ instead of $\lambda$.

  \newslide
\minisec{Historical Remark:}
%\normalfont\footnotesize%
The Lorenz condition and gauge
are named in honor of the Danish theoretical physicist\\
%
\highlighttext{Ludwig Valentin Lorenz} (1829--1891),\\
%
who has first published it in 1867~\cite{Lorenz/1867}
(see also~\cite[pp.\ 268-269, 291]{Whittaker-History});
however this condition was first introduced in lectures by Bernhard
G.~W.~Riemann in 1861 as pointed in~\cite[p.~291]{Whittaker-History}. It
should be noted that the \emph{Lorenz} condition/gauge is quite often
erroneously referred to as the Loren\emph{t}z condition/gauge after the name of
the Dutch theoretical physicist Hendrik Antoon Lorentz (1853--1928) as, e.g.,
in~\cite[p.~18]{Roman-QFT} and in~\cite[p.~45]{Gockeler&Schucker}.%

\end{slide}
%			END OF SECTION~\ref{Sect4}
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%			BEGINNING OF SECTION~\ref{Sect5}
\begin{slide}
\section{Normal and inertial frames}
\pageTransitionDissolve% Dissolve


	Comparing~\eref{4-5n.18} with~\eref{4-5.6}, we get
(in this section, we assume the Greek indices to run over the range
0, 1, 2, 3.
)
	\begin{equation}	\label{4-5n.23}
F_{\mu\nu} = R_{\mu\nu}(-A_0,-A_1,-A_2,-A_3) .
	\end{equation}
Thus, the electromagnetic field tensor  $F$ is completely responsible for the
existence of frames normal for $L$ (theorems~\ref{4-Thm5.1}
and~\ref{4-Thm5.2}). For example, if $U$ is an open set, frames normal on
$U\subseteq M$ for $L$ exist iff $F|_U=0$, \ie if electromagnetic field
is missing on $U$.
Also, if $N$ is a submanifold of $M$, frames normal on $U$ for $L$ exist iff
in the special coordinates $\{x^\mu\}$, described in theorem~\ref{4-Thm5.2},
is valid $F_{\alpha\beta}|_U=0$ for  $\alpha,\beta=1,\dots,\dim N$.

%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionReplace% The default simple replace


In the context of theorem~\ref{4-Thm3.1}, we can say
that an electromagnetic field admits frames normal on $U\subseteq M$ iff
the linear transport $L$ corresponding to it is
\emph{path\ndash independent} on
$U$ (along paths lying entirely in $U$). Thus, if $L$ is path\ndash dependent
on $U$, the field does not admit frames normal on $U$. This important result
is the classical analogue of a quantum effect, know as the Aharonov\ndash
Bohm effect~\cite{Aharonov&Bohm, Bernstein&Phillips}, whose essence is that
the electromagnetic potentials directly, not only through the field tensor
$F$, can give rise to observable physical results.
%--------------------------------------
\newslide
%--------------------------------------

	\minisec{Physical meaning of the normal frames}
% corresponding to a given
% electromagnetic field which is described, as pointed above, via a linear
% transport $L$ along paths in a line vector bundle over the space\ndash time
% $M$.

	Suppose $L$ is Euclidean on a neighborhood $U\subseteq M$. As a
consequence of~\eref{4-5n.23} and theorem~\ref{4-Thm5.1}, we have $F|_U=\od
A|_U=0$, \ie on $U$ the electromagnetic field strength vanishes and hence the
field is a pure gauge on $U$,
	\begin{equation}	\label{4-5n.24}
A_\mu|_U = \frac{\pd f_0}{\pd x^\mu}\Big|_U
	\end{equation}
for some  $C^1$ function $f_0$ defined on an open set containing $U$ or equal
to it.
In a frame $\{e'\}$ normal
on $U$ for $L$ vanish the 2\ndash index coefficients of $L$ along any path
$\gamma$ in $U$:
	\begin{equation}	\label{4-5n.25}
\Mat{\Gamma}'(s;\gamma) = A'_\mu(\gamma(s)) \dot\gamma^\mu(s) = 0 .
	\end{equation}
% for every $\gamma\colon J\to U$ and $s\in J$.


%--------------------------------------
\newslide
%--------------------------------------

   Any transformation~\eref{4-5n.20} with
% 	\begin{equation}	\label{4-5n.26}
\(
\lambda = -f_0
\)
% 	\end{equation}
transforms $A_\mu$ into $A'_\mu$ such that
	\begin{equation}	\label{4-5n.27}
A'_\mu|_U = 0
	\end{equation}
(irrespectively of the frames $\{E_\mu\}$ and $\{E'_\mu\}$.)
%
Hence, the one\ndash vector frame
$\{e'=\e^{-f_0}e\}$ in the bundle space $E$ is normal for $L$ on $U$.
Therefore, in the frame $\{e'\}$, vanish the 2\ndash index
and \emph{3\ndash index} coefficients of $L$, \ie $\{e'\}$ is a
\emph{strong normal} frame.
One can
verify,
\emph{%
all frames strong normal on a neighborhood $U$ for $L$ are obtainable from
$\{e'\}$ by multiplying its vector $e'$ by a function $f$ such that
$\frac{\pd f}{\pd x^\mu}\bigr|_U=0$%
},
\ie they are $\{b\e^{-f_0}e\}$ with $b\in\mathbb{R}\backslash\{0\}$ as  $U$ is
a neighborhood. Thus, every frame normal on a neighborhood $U$ for $L$ is
strong normal on U for $L$ and vice versa.

%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionSplitHO% Split Horizontally Outside


	A frame (of reference) in the bundle space in which~\eref{4-5n.27}
holds on a subset $U\subseteq M$, will be called \emph{inertial on $U$ for the
electromagnetic field} considered. In other words, the frames inertial on $U$
for a given electromagnetic field are the ones in which its potentials vanish
on $U$. Thus, every frame inertial on $U$ is strong normal on it and vice
versa.

	So, in a frame inertial on $U\subseteq M$ for an electromagnetic
field it is not only a pure gauge, but in such a frame its potentials vanish
on $U$.

%--------------------------------------
\newslide
%--------------------------------------
 \pageTransitionSplitHI% Split Horizontally Outside

 Relying on known
results~\cite{bp-Frames-n+point,bp-Frames-path,bp-Frames-general}, we can
assert
\emph{the existence of frames inertial at a single point and/or along
paths without self\ndash intersections for every electromagnetic field}, while
on submanifolds of dimension not less than two such frames exist only as an
exception if (and only if) some additional conditions are satisfied, \ie for
some particular types of electromagnetic fields.

%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionSplitVO% Split Vertically Outside

\minisec{Links with the (strong) equivalence principle}

	In~\cite{bp-PE-P?}
was demonstrated that the (ordinary strong)
\highlighttext{equivalence principle is a
provable theorem}
 and the inertial frames in a gravity theory based
on a linear connection (or other derivation) are the frames normal for it.
Such
frames are called \emph{inertial for the gravitational field} under
consideration.

	Let there be given a physical system consisting of pure
or, possibly, interacting gravitational and electromagnetic fields which are
described via, respectively, a linear connection $\nabla$ in the tangent
bundle $(T(M),\pi_T,M)$ (or the tensor algebra) over the space\ndash time
$M$ and a linear transport along paths in a 1\ndash dimensional vector bundle
$(E,\pi_E,M)$ over $M$.


%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionSplitVI% Split Vertically Inside

    \begin{itemize}
\item
The frames inertial
for an electromagnetic field, if any, in the bundle space $E$ are completely
independent of any frame in the bundle space $T(M)$ tangent to $M$.
\item
The frames inertial for the gravity field, \ie the ones normal
for $\nabla$, if any, are frames in $T(M)$ and have nothing in common with the
frames in $E$, in particular with the frames normal for $L$, if any.
    \end{itemize}

\pause
So, if a frame $\{E_\mu\}$ in $T(M)$ is inertial on
$U\subseteq M$ for the gravity field and a frame $\{e\}$ in $E$ inertial on
$U$ for the electromagnetic field, the frame
$\{e\times E_\mu\}=\{(e,E_\mu)\}$ in the bundle space of
$(E\times T(M),\pi_E\times\pi_T,M\times M)$
over $M\times M$ can be called
simply \emph{inertial} on $U$ (for the system of gravity and electromagnetic
fields).


%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionBlindsH% Horizontal Blinds


Thus, in an inertial frame, if any, the potentials of both, gravity
and electromagnetic, fields vanish.
We can
assert the
\highlighttext{existence of inertial frames at every single space\ndash time
point and/or along every path without self\ndash intersections in it.}
On
submanifolds of dimension higher than one, inertial frames exist only for
some exceptional configurations of the fields which can be described on
the base of the results in the cited works.

\end{slide}
%			END OF SECTION~\ref{Sect5}
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\begin{slide}

\pageDuration{9}

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 \bibliographystyle{unsrt}
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\end{slide}

\pagetransitionDissolve

    \begin{slide}
The present report is available on the Internet as:\\
\begin{itemize}
\item
\LaTeX2e file at the following URL:\\
\href{http://theo.inrne.bas.bg/$\sim$bozho/publ/Publication_Files/nf-ltp-lb_CSVF2006_report.tex}
     {http://theo.inrne.bas.bg/$\sim$bozho/publ/}\\
\href{http://theo.inrne.bas.bg/$\sim$bozho/publ/Publication_Files/nf-ltp-lb_CSVF2006_report.tex}
     {Publication$\_$Files/nf-ltp-lb$\_$CSVF2006$\_$report.tex}

\item
Acrobat pdf file at the following URL:\\
\href{http://theo.inrne.bas.bg/$\sim$bozho/publ/Publication_Files/nf-ltp-lb_CSVF2006_report.pdf}
     {http://theo.inrne.bas.bg/$\sim$bozho/publ/}\\
\href{http://theo.inrne.bas.bg/$\sim$bozho/publ/Publication_Files/nf-ltp-lb_CSVF2006_report.tex}
     {Publication$\_$Files/nf-ltp-lb$\_$CSVF2006$\_$report.pdf}
\end{itemize}
    \end{slide}





\end{document}
