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A Riemannian metric $g$ is symmetric because its value $g_p$ at each point $p$ is an inner product on the tangent space $T_pM$. Inner products are symmetric, so $g_p(v,w) = g_p(w,v)$, which implies $g(X,Y) = g(Y,X)$ for vector fields.

Conditions: $g$ is a Riemannian metric.; $g_p$ is an inner product on $T_pM$ for all $p \in M$.

A Riemannian metric $g$ is symmetric because its value $g_p$ at each point $p$ is an inner product on the tangent space $T_pM$. Inner products are symmetric, so $g_p(v,w) = g_p(w,v)$, which implies $g(X,Y) = g(Y,X)$ for vector fields.

Conditions: $g$ is a Riemannian metric.; $g_p$ is an inner product on $T_pM$ for all $p \in M$.