What is the formal definition of a Riemannian metric on a smooth manifold?

A covariant 2-tensor field is linear over the ring of smooth functions. If $X, Y$ are vector fields and $f, h$ are smooth functions, then $g(fX, hY) = f h g(X,Y)$.

Conditions: $g$ is a covariant 2-tensor field.; $X, Y$ are vector fields.; $f, h$ are smooth functions.

Positive definite means that for any tangent vector $v \in T_pM$, $g_p(v,v) \ge 0$, and $g_p(v,v) = 0$ if and only if $v = 0$. This ensures the metric defines a valid norm.

Conditions: $g$ is a Riemannian metric.; $v \in T_pM$ is a tangent vector.

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 covariant tensor field takes vectors as inputs and outputs a scalar number. In the context of a Riemannian metric, it operates on pairs of vectors from the tangent space at a given point.

Conditions: The tensor field is covariant.; Inputs are vectors from the tangent space.; Output is a scalar.