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\date{12 August 2006\footnote{from a news article [1]}}
\author{Ulrich~\textsc{Bruchholz}\footnote{Dipl.-Ing.
\textsl{Ulrich Bruchholz},~http://www.bruchholz-acoustics.de}}
\title{\textsc{Riemann}ian Geometry of the Space-Time}

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Investigating the properties of diverse manifolds, \textsc{Donaldson} [2]
as well as \textsc{Freedman} [3] determined special properties of the
four-manifold. In that, \textsc{Donaldson}'s work is based on the work
of a group around \textsc{Atiyah} (\textsc{Donaldson} [4]), who include
the \textsc{Yang-Mills} equations. These involve ``exotic smooth
structures'' of the four-manifold.

This step has been done with the belief in the physical relevance
of quantum field theory. Physicists take the \textsc{Yang-Mills} equations
as generalizations of \textsc{Maxwell}'s equations. However, the
author could demonstrate [5], that \textsc{Maxwell}'s equations
go into the system of source-free \textsc{Einstein-Maxwell} equations
{\large
\begin{equation}
R_{ik} = \kappa \cdot ( \frac{1}{4}~g_{ik} F_{ab} F^{ab} - F_{ia} F_k{}^a )
\qquad ,
\end{equation}
\begin{equation}
        F^{ia}{}_{;a} = 0 \qquad ,
\end{equation}
\begin{equation}
        F_{ik} = A_{i,k} - A_{k,i} \qquad .
\end{equation}}
\normalsize
Physicists call that an electrovacuum. As well, the material quantities
like mass, spin, charge, magnetic momentum are integration constants
of above system of PDE. Physicists always claim that this step would
disregard the quantum phenomena. Strangely enough, the values of the
integration constants for most stable solutions from numerical
simulations are just the known particle numbers [5].

Equ. (1) to (3)
involve a special kind of \textsc{Riemann}ian geometry of the four-manifold
of signature (+,+,+,--), what is explained as follows. As well, the
derivation follows in general that of the \textsc{Ricci} main directions
as done by \textsc{Eisenhart} [6]. Unlike all other manifolds, the
results for the four-manifold involve two main surfaces instead of
four main directions.

The \textsc{Ricci} main directions (written in terms according to \textsc{Eisenhart})
follow from
{\large
\begin{equation}
        \mathsf{det} | R_{ik} + \rho g_{ik} | = 0
\end{equation}}
\normalsize
with the solutions
{\large
\begin{equation}
    \rho_{|1} = \rho_{|4} = +\rho_o \qquad ,
    \qquad \qquad  \rho_{|2} = \rho_{|3} = -\rho_o
\end{equation}}
\normalsize
with
{\large
\begin{equation}
    \rho_o{}^2 = R_1{}^a R^1{}_a = R_2{}^a R^2{}_a = R_3{}^a R^3{}_a = R_4{}^a R^4{}_a
\qquad .
\end{equation}}
\normalsize
Characteristical are the two double-roots, that means: There are
two dual surfaces of the congruences
$ e_{|1}{}^i e_{|4}{}^k - e_{|1}{}^k e_{|4}{}^i $
and $ e_{|2}{}^i e_{|3}{}^k - e_{|2}{}^k e_{|3}{}^i $
with extreme mean \textsc{Riemann}ian curvature.
$e_{|1} ... e_{|4}$ are the vectors of an orthogonal quadrupel
in those ``main surfaces''.

With it we get
{\large
\begin{equation}
g_{ik} =   e_{|1\_i} e_{|1\_k} + e_{|2\_i} e_{|2\_k} + e_{|3\_i} e_{|3\_k} - e_{|4\_i} e_{|4\_k}
\qquad ,
\end{equation}
\begin{equation}
\frac{R_{ik}}{\rho_o}
     = - e_{|1\_i} e_{|1\_k} + e_{|2\_i} e_{|2\_k} + e_{|3\_i} e_{|3\_k} + e_{|4\_i} e_{|4\_k}
\qquad .
\end{equation}}
\normalsize
If we set
{\large
\begin{equation}
        c_{|ik} = -c_{|ki} = F_{ab} e_{|i}{}^a e_{|k}{}^b
\end{equation}}
\normalsize
follows
{\large
\begin{equation}
        -\kappa \Bigl( (c_{|23})^2 + (c_{|14})^2 \Bigr) = 2 \rho_o \qquad ,
\end{equation}
\begin{equation}
        c_{|12} = c_{|34} = c_{|13} = c_{|24} = 0 \qquad .
\end{equation}}
\normalsize
With it, the field tensor
{\large
\begin{equation}
        F_{ik} = - c_{|14} (e_{|1\_i} e_{|4\_k} - e_{|1\_k} e_{|4\_i})
               + c_{|23} (e_{|2\_i} e_{|3\_k} - e_{|2\_k} e_{|3\_i})
\end{equation}}
\normalsize
is performed from the main surfaces !

With it, these surfaces depict the electromagnetic fields same way
like the curvature vector does it with acceleration + gravitation.
Both quantities are parameters of the curve as described by
each body in the space-time. They accompany this curve.

It were a great task for mathematicians to discuss the properties
of this special manifold. We have to take notice of geometric
boundaries.\\

%\newpage
\begin{center}\textbf{References}\\ \end{center}
{}[1] From: Ulrich Bruchholz\\
Message-ID: \textless9j71vm\$lp8mr\$1@ID-28113.news.dfncis.de\textgreater\\
Newsgroups: sci.physics.research\\
Subject: Ricci Main Directions in an EM Vacuum\\
Date: 19 Jul 2001 16:30:46 GMT\\
Approved: spr@rosencrantz.stcloudstate.edu (sci.physics.research)\\
\\
{}[2] http://www.britannica.com/eb/article-9096161\\
\\
{}[3] http://www-history.mcs.st-andrews.ac.uk/Biographies/Freedman.html\\
\\
{}[4] \textsc{Donaldson}, S.: Geometry in Oxford C.1980-85,\\
Asian J. Math. \textbf{3}, 43, 1999.\\
\\
{}[5] http://bruchholz.psf.net\\
http://UlrichBruchholz.homepage.t-online.de\\
with further references.\\
\\
{}[6] \textsc{Eisenhart}, L.P.: Riemannian Geometry,\\
Princeton university press.
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