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
applies.
The metric is thus covariant constant with respect to the metric relationship. From this quality it follows for everyone
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.
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
so with the notation
- for each the equation holds,
- for each and for all the associative law applies and
- for all the mapping is bijective .
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 ).