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\date{12 January 2007}
\author{Ulrich~\textsc{Bruchholz}\footnote{Dipl.-Ing.
\textsl{Ulrich Bruchholz},~http://www.bruchholz-acoustics.de}}
\title{Special Parameters of a Curve in~the~Four-Manifold}

\begin{document}
\sloppypar
\maketitle
\begin{abstract}
It is demonstrated that a four-manifold at least of signature
(+,+,+,--) is characterized by two special parameters of any
curve\footnote{under consideration of the peculiarities from the
signature} in it, which reflect the properties of the manifold.
That are the curvature vector and a special asymmetric tensor
of second rank. They accompany the curve.
\end{abstract}

Any curve in a $n$-manifold is accompanied by a special orthogonal
ennuple or $n$-bein. That are $n$ mutually orthogonal unit vectors.
The vectors of the accompanying $n$-bein are determined by the
generalized \textsc{Frenet} formulae according to \textsc{Blaschke}
(\textsc{Eisenhart}~[1]).

The first vector is the tangent vector of the curve
{\large \begin{equation}
t^{i} = \frac{\mathsf{d}x^i}{\mathsf{d}s} \quad .
\end{equation}}
The second vector is the main normal 
{\large \begin{equation}
n^{i} = \rho~t^{a} t^i{}_{;a} \quad ,
\end{equation}}
in which $\rho$ means the curvature radius.\\
The most important curve parameter might be the curvature vector
{\large \begin{equation}
k^i = \frac{n^i}{\rho} = t^{a} t^i{}_{;a} =
\frac{\mathsf{d}^{2}x^i}{\mathsf{d}s^2} + \{_a{}^i{}_b\}~t^{a} t^{b} \quad .
\end{equation}}

Are there other curve parameters ?\\
The four-manifold opens the possibility of
parameters that are performed from the congruences of two dual surfaces
instead of single vectors. A parameter analogous the curvature vector
were an asymmetric tensor of second rank, and must be written with
these surfaces. 

This asymmetric tensor is expressible from a vector potential
{\large \begin{equation}
        F_{ik} = A_{i,k} - A_{k,i} \qquad .
\end{equation}}
If the divergences vanish
{\large \begin{equation}
        F^{ia}{}_{;a} = 0 \qquad ,
\end{equation}}
it exists a relation to the \textsc{Ricci} tensor
{\large
\begin{equation}
R_{ik} = \frac{1}{4}~g_{ik} F_{ab} F^{ab} - F_{ia} F_k{}^a \qquad .
\end{equation}}
(A constant part of the \textsc{Riemann}ian curvatures would not
disturb the derivation, but the description were not more clear.)

Equ. (4) to (6)
involve a special kind of \textsc{Riemann}ian geometry of the four-manifold
of signature (+,+,+,--)\footnote{It were to investigate if this signature
is necessary.}. This geometry leads indeed to special dual
surfaces, which perform just mentioned asymmetric tensor. As well, the
derivation follows in general that of the \textsc{Ricci} main directions
as done by \textsc{Eisenhart} [1]. 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}}
with the solutions
{\large
\begin{equation}
    \rho_{|1} = \rho_{|4} = +\rho_o \qquad ,
    \qquad \qquad  \rho_{|2} = \rho_{|3} = -\rho_o
\end{equation}}
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}}
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 (vierbein)
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}}
If we set
{\large
\begin{equation}
        c_{|ik} = -c_{|ki} = F_{ab} e_{|i}{}^a e_{|k}{}^b
\end{equation}}
follows
{\large
\begin{equation}
        - \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}}
With it, the asymmetric 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}}
is performed from the main surfaces !\\
That means, this tensor is indeed a curve parameter too.
It accompanies the curve like the curvature vector does it.\\
Both curve parameters, the curvature vector and this tensor,
reflect the properties of the four-manifold.\\
\\ 
\\
%\newpage
\begin{center}\textbf{References}\\ \end{center}
{}[1] \textsc{Eisenhart}, L.P.: Riemannian Geometry,\\
Princeton university press.
\end{document}
