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A Riemannian metric $g$ on a smooth manifold $M$ is a smooth covariant 2-tensor field such that for each point $p \in M$, the value $g_p$ is an inner product on the tangent space $T_pM$. It is symmetric and positive definite.

Conditions: $M$ is a smooth manifold.; $g$ is a smooth covariant 2-tensor field.; For each $p \in M$, $g_p$ is an inner product on $T_pM$.; $g_p(v,v) \ge 0$ for all $v \in T_pM$.; $g_p(v,v) = 0$ if and only if $v = 0$.

A Riemannian metric $g$ on a smooth manifold $M$ is a smooth covariant 2-tensor field such that for each point $p \in M$, the value $g_p$ is an inner product on the tangent space $T_pM$. It is symmetric and positive definite.

Conditions: $M$ is a smooth manifold.; $g$ is a smooth covariant 2-tensor field.; For each $p \in M$, $g_p$ is an inner product on $T_pM$.; $g_p(v,v) \ge 0$ for all $v \in T_pM$.; $g_p(v,v) = 0$ if and only if $v = 0$.

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.

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.