Given a Riemannian manifold \((M,g)\). A connection \(\nabla\) on \(TM\) that is compatible with \(g\) and symmetric is called Levi-Civita Connection.
Remarks
- On every Riemannian manifold exists a unique Levi-Civita connection. This is the so called Fundamental theorem of Riemannian geometry [1, Theorem 5.10]. It is therefore the canonical choice for a connection.
Links Link to heading
- Christoffel symbols
- induced connection is compatible with the inner product on tensor bundles
- Levi-Civita connection commutes with musical isomorphisms
References Link to heading
- J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.