Metric connection

A metric relationship or a relationship compatible with the metric is a mathematical object from differential geometry . It is a special case of a relationship .

definition

Let be a Riemannian manifold and be a vector bundle with (induced) metric . A connection on is called a metric connection, if for all sections ${\displaystyle (M,{\tilde {g}})}$ ${\displaystyle (E\to M,g)}$ ${\displaystyle g}$ ${\displaystyle \nabla }$ ${\displaystyle E}$ ${\displaystyle X,Y,Z\in \Gamma (E)}$

${\displaystyle \nabla _{X}(g(Y,Z))=0}$

applies.

The metric is thus covariant constant with respect to the metric relationship. From this quality it follows for everyone ${\displaystyle X,Y,Z\in \Gamma (E)}$

${\displaystyle X(g(Y,Z))=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z).}$

Examples

The best-known example of a metric relationship is the Levi-Civita relationship . In this case the vector bundle is the tangential bundle an with the Riemannian metric of . Since there is exactly one Levi-Civita connection for every Riemannian manifold, there is in particular at least one metric connection on a Riemannian manifold. ${\displaystyle TM}$ ${\displaystyle M}$ ${\displaystyle M}$

Affine space

Let be a vector bundle with metric then the set of metric connections is modeled on a non-empty affine space with the (vector valued) 1-forms from d. i.e. there is a picture ${\displaystyle (E\to M,g)}$ ${\displaystyle g,}$ ${\displaystyle X}$ ${\displaystyle E}$ ${\displaystyle {\mathcal {A}}^{1}(M,\operatorname {End} (E)),}$

${\displaystyle l:{\mathcal {A}}^{1}(M,\operatorname {End} (E))\times X\to X,}$

so with the notation ${\displaystyle \omega +\nabla :=l(\omega ,\nabla )}$

1. for each the equation holds, ${\displaystyle \nabla \in X}$ ${\displaystyle 0+\nabla =\nabla }$
2. for each and for all the associative law applies and ${\displaystyle \omega ,\nu \in {\mathcal {A}}^{1}(M,\operatorname {End} (E))}$ ${\displaystyle \nabla \in E}$ ${\displaystyle (\omega +\nu )+\nabla =\omega +(\nu +\nabla )}$
3. for all the mapping is bijective . ${\displaystyle \nabla \in X}$ ${\displaystyle \omega \mapsto \omega +\nabla }$

literature

• John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 .
• Manfredo Perdigão do Carmo: Riemannian Geometry. Birkhäuser, Boston et al. 1992, ISBN 0-8176-3490-8 .
• Nicole Berlin, Ezra Getzler , Michèle Vergne : Heat Kernels and Dirac Operators (= Fundamentals of Mathematical Sciences 298). Corrected 2nd printing. Springer, Berlin et al. 1996, ISBN 3-540-53340-0 .
• Ü. Lumiste: Metric connection . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).