Riemannian Manifold

Definition

A Riemannian manifold is a smooth Topological Manifold together with a Riemannian metric. Many geometric notions such as distance, angles, length, volume, and curvature are defined on a Riemannian manifold.

Let $\mathcal{M}$ be a Smooth Manifold. For each point $p\in \mathcal{M}$, there is an associated vector space $T_p(\mathcal{M})$ called Tangent Space of $\mathcal{M}$ at $p$. Each Tangent Space $T_p(\mathcal{M})$ equipped with an Inner product $g_p:T_p(\mathcal{M})\times T_p(\mathcal{M})\to \mathbb{R}$ defined by the basis of the tangent space. $g_{n\times n}=(g_{ij}),g_{ij}{|}_p:=⟨{\partial }_i,{\partial }_j⟩$ Where $⟨\cdot ,\cdot ⟩$ is a standard inner product of the ambient space (Euclidean inner product).

The collection of inner products $g=\{g_p|p\in \mathcal{M}\}$ is a Riemannian metric on $\mathcal{M}$, and the pair of the manifold $\mathcal{M}$ and the Riemannian metric $g$ defines a Riemannian manifold. $(\mathcal{M},g)$