📂Geometry

Coexistence Compatible Connection

Definition¹

Let’s assume that an affine connection $M$ and a Riemannian metric $\nabla$ are given on a differentiable manifold. For all differentiable curves $g$, if any parallel vector fields $c$ along any two $g$ satisfy $c$, then the connection $M$ is said to be compatible with the metric $\nabla$.

Explanation

The following corollary is sometimes stated as the definition of compatibility. The definition given above is easy to conceptualize because of the condition $c$, but it is difficult to use in practice. On the other hand, the condition in the corollary is practically useful for formulating equations though its meaning may not be immediately apparent.

Theorem

Let’s consider $P, P^{\prime}$ a Riemannian manifold. A necessary and sufficient condition for the connection $M$ to be compatible with the metric $\nabla$ is that for all vector fields $g(P, P^{\prime}) = \text{constant}$ following a differentiable curve $\nabla$, the following holds:

$$\dfrac{d }{d t}g(V,W) = g\left( \dfrac{DV}{dt}, W \right) + g\left( V, \dfrac{DW}{dt} \right),\quad t\in I$$

Corollary

A necessary and sufficient condition for a connection $M$ on a Riemannian manifold $P, P^{\prime}$ to be compatible with the metric $\nabla$ is the following:

$$X g(Y,Z) = g\left( \nabla_{X}Y, X \right) + g\left(Y, \nabla_{X}Z \right),\quad X,Y,Z \in \mathfrak{X}(M)$$

Here, $g$ represents the collection of vector fields on $M$.