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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.

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.