Levi-Civita connection
On any Riemannian manifold ⟨M,g⟩, there is a unique torsion-free affine connection ∇ on the tangent bundle of M such that the covariant derivative of the metric tensor g is zero, i.e. g is covariantly constant. This condition can be also be expressed in terms of the inner product operation ⟨,⟩:TM×TM→ℝ induced by g as follows: For all vector fields X,Y,Z∈TM, one has
X(⟨Y,Z⟩)=⟨∇XY,Z⟩+⟨Y,∇XZ⟩ |
and
∇XY-∇YX=[X,Y] |
This connection is called the Levi-Civita connection.
In local coordinates {x1,…,xn}, the Christoffel symbols (http://planetmath.org/Connection) Γijk are determined by
giℓΓijk=12(∂gjℓ∂xk+∂gkℓ∂xj-∂gjk∂xℓ). |
Title | Levi-Civita connection |
---|---|
Canonical name | LeviCivitaConnection |
Date of creation | 2013-03-22 13:59:25 |
Last modified on | 2013-03-22 13:59:25 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 7 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 53B05 |