Riemannian manifold

Definition

A Riemannian manifold $M$ is a manifold with smooth bundle metric as a Riemannian metric $g$ where $g$ defined the following. For each $p\in M$,

$$g_p:T_{p}M\times T_{p}M\to \mathbb{R}.$$

1. Bilinear.
2. Symmetric.
3. Positive-definite.