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?

  • $\begingroup$ 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. $\endgroup$ – Moishe Kohan Dec 27 '18 at 4:34

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