Definition of Riemannian Metric

Question

Let $(M, g)$ be a Riemannian manifold. Standard definitions of a Riemannian metric $g$ states that $g$ specifies a symmetric, bilinear, positive definite form on each tangent space $T_{p}M$ that varies smoothly with $p$. Why do we not require $g$ to satisfy the triangle inequality in our definition of a Riemannian metric? Is it simply because all we really need to study geometry is an inner product structure which induces a distance function that satisfies the usual definition of a metric?