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Riemannian Geometry (Definition)

The study of Riemannian manifolds, which are smooth manifolds equipped with a Riemannian metric (an inner product on the tangent space).
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The statement of the theorem

Let MM be an nn-dimensional smooth manifold, equipped with an atlas A={(Ui,ϕi)}i=1k\text{A} = \{\text{(U}_i, \phi_i\text{)}\text{}\}_{i=1}^{k}. A Riemannian metric gg on MM is a smooth section of the bundle Sym2(T∗M)\text{Sym}^2(T^*M), which assigns to every point p∈Mp \in M a positive-definite inner product gpg_p on the tangent space TpMT_p M. Formally, gg must satisfy the following conditions:\n\n1. **Smoothness:** For any local chart ϕi:Ui→Rn\phi_i: U_i \to \mathbb{R}^n, the components gij=g(∂∂xi,∂∂xj)g_{ij} = g(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}) must be smooth functions on UiU_i.\n2. **Positive Definiteness:** For all p∈Mp \in M and any non-zero tangent vector v∈TpMv \in T_p M, the metric must satisfy gp(v,v)>0g_p(v, v) > 0.\n\nThus, a Riemannian manifold is the pair (M,g)(M, g), where gg is a smooth, positive-definite, symmetric (0,2)(0, 2)-tensor field on MM. The associated Riemannian volume element is volg=det⁡(gij) dx1∧⋯∧dxn\text{vol}_g = \sqrt{\det(g_{ij})}\ dx^1 \wedge \dots \wedge dx^n.