Riemannian manifold - Knowlpedia

Let $M$ be a finite-dimensional smooth manifold. A Riemannian metric on $M$ is a smooth symmetric covariant $2$ -tensor $g$ such that $g_p$ is a positive-definite inner product on $T_pM$ for every $p\in M$ . Equivalently, $g$ is a bundle metric on the tangent bundle $TM$ . A Riemannian manifold is a pair $(M,g)$ . In local coordinates, $g=g_{ij}\,dx^i\otimes dx^j$ , where the matrix $(g_{ij}(p))$ is symmetric positive definite at every point and each coefficient $g_{ij}$ is smooth.

Metric and volume constructions

The metric assigns each tangent vector a length $\lVert v\rVert_g=\sqrt{g(v,v)}$ and each piecewise smooth curve $\gamma$ a length

L_g(\gamma)=\int \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt.

Taking the infimum of curve lengths gives the Riemannian distance on each connected component. The metric also identifies $TM$ with the cotangent bundle and determines a canonical volume density; an orientation turns this density into a volume form.

Canonical connection and curvature

Every Riemannian metric has a unique torsion-free, metric-compatible connection, the Levi–Civita connection. It determines geodesics, parallel transport, and curvature. These structures depend on derivatives of $g$ , whereas lengths and angles are pointwise data.

Examples and scope

The Euclidean metric $\sum_i dx^i\otimes dx^i$ is the standard example on $\mathbb R^n$ . An immersion into Euclidean space pulls this metric back to a Riemannian metric when its differential is injective. Positive definiteness distinguishes Riemannian metrics from pseudo-Riemannian metrics, whose nondegenerate symmetric forms may have mixed signature. Smoothness is essential: an arbitrary choice of an inner product on each tangent space does not necessarily define a Riemannian metric.

References

John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Graduate Texts in Mathematics 176, Springer, 2018. Publisher record. Relevant: Chapters 2–5.
Manfredo P. do Carmo, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser, 1992. Publisher record. Relevant: Chapters 0–3.