# Equivalence of two ways of defining a Riemannian metric

Definition 1 A Riemannian metric on a smooth manifold is smooth family of inner products on the tangent spaces of M. So g is Riemannian metric if it assigns to each point $$p \in M$$ a positive definite symmetric bilinear form on $$T_p M$$,

$$g_p: T_p M \times T_p M \rightarrow \mathbb{R}$$

with smoothness referring to the requirement that the function

$$p \mapsto g_p(X_p, Y_p)$$

must be smooth for any locally defined vector fields X,Y in M.

Definition 2 A Riemannian metric on a smooth manifold X is a smooth section of $$S^2 T^* X \subset \otimes^2 T^*X$$ such that $$\left.f\right|_X \in S^2 T^*_xX$$ is a positive definite quadratic form on $$T_xX$$.

Why are these two definitions equivalent?

• Just write down both definitions in local coordinates; then you will see that the two notions are equivalent. Keep in mind that to verify 1 it suffices to check it for (local) coordinate vector fields. – Moishe Kohan Dec 27 '18 at 4:34