Differentialgeometrie/Gemischte Definitionsabfrage/1/Aufgabe/Lösung

${\displaystyle {}r={\frac {1}{\vert {\gamma _{1}'(t_{0})\gamma _{2}^{\prime \prime }(t_{0})-\gamma _{2}'(t_{0})\gamma _{1}^{\prime \prime }(t_{0})}\vert }}\,}$

und dem Mittelpunkt

${\displaystyle {}M=\gamma (t_{0})+{\frac {1}{\gamma _{1}'(t_{0})\gamma _{2}^{\prime \prime }(t_{0})-\gamma _{2}'(t_{0})\gamma _{1}^{\prime \prime }(t_{0})}}{\begin{pmatrix}-\gamma _{2}'(t_{0})\\\gamma _{1}'(t_{0})\end{pmatrix}}\,}$

den Krümmungskreis zu ${\displaystyle {}\gamma }$ in ${\displaystyle {}t_{0}}$.

${\displaystyle {}M\subseteq N}$

${\displaystyle {}P\in M}$

${\displaystyle {}P\in W\subseteq N}$

${\displaystyle {}\theta \colon W\rightarrow W'}$

${\displaystyle {}W'\subseteq \mathbb {R} ^{n}}$

${\displaystyle {}M\cap W=\theta ^{-1}((\mathbb {R} ^{m}\times \{0\})\cap W')\,.}$

${\displaystyle {}M}$

${\displaystyle {}\partial M={\left\{x\in M\mid \alpha _{i}(x)\in \partial H\cap V_{i}{\text{ für }}{\text{ein }}i\in I\right\}}\,}$

definiert, wobei ${\displaystyle {}\alpha _{i}}$ Karten sind.

${\displaystyle {}{\left(g^{kl}\right)}}$

${\displaystyle {}{\left(g_{ij}\right)}}$

${\displaystyle {}U}$

${\displaystyle {}\Gamma _{ij}^{k}={\frac {1}{2}}\sum _{l=1}^{n}g^{kl}(\partial _{i}g_{jl}+\partial _{j}g_{il}-\partial _{l}g_{ij})\,}$

die Christoffelsymbole für den Levi-Civita-Zusammenhang.