Riemannian Geometry as a Tangent-Bundle Connection

Riemannian geometry — metrics, geodesics, the Riemann curvature tensor, general relativity — is the special case of the bundle machinery of this folder applied to the tangent bundle. Seeing it this way unifies gauge theory and gravity: both are connections with curvature, differing only in which bundle. This page is the bridge to geometry/curvature.md and general relativity; it builds on connections.md and curvature.md.

The Levi-Civita connection

A Riemannian metric $g$ is a smooth, positive-definite inner product on each tangent space — equivalently, a reduction of the frame bundle's structure group from $G L (n)$ to $O (n)$ (fiber-bundles.md). On the tangent bundle $TM$ there is a distinguished connection:

Fundamental theorem of Riemannian geometry. There is a unique connection on $TM$ that is (i) metric-compatible ( $\nabla g = 0$ ) and (ii) torsion-free ( $\nabla_{X} Y - \nabla_{Y} X = [X, Y]$ ): the Levi-Civita connection.

Its coefficients are the Christoffel symbols $Γ_{μν}^{ρ} = \frac{1}{2} g^{ρ σ} (\partial_{μ} g_{ν σ} + \partial_{ν} g_{μ σ} - \partial_{σ} g_{μν}),$ which play exactly the role of the gauge potential $A_{μ}$ of connections.md, with structure group $O (n)$ (or the Lorentz group $O (1, 3)$ in relativity). Parallel transport, geodesics ( $\nabla_{\overset{γ}{˙}} \overset{γ}{˙} = 0$ , "straightest lines"), and the covariant derivative are the tangent-bundle instances of the general definitions.

Riemann curvature is bundle curvature

The Riemann curvature tensor is the curvature $F$ of the Levi-Civita connection: $[\nabla_{μ}, \nabla_{ν}] V^{ρ} = R^{ρ}_{σ μν} V^{σ},$ the same "commutator of covariant derivatives" that defines gauge curvature in curvature.md, now with tangent-space (Lorentz) indices instead of gauge indices. The Bianchi identity $\nabla_{[λ} R_{μν]}^{ρ}_{σ} = 0$ is the tangent-bundle version of $d_{\nabla} F = 0$ , and (contracted) gives the conservation $\nabla_{μ} G^{μν} = 0$ of the Einstein tensor — the reason energy–momentum is conserved in general relativity.

The gauge theory / gravity dictionary

Gauge theory	Gravity
structure group $G$ (internal)	Lorentz group $O (1, 3)$ (frame)
gauge potential $A_{μ}$	spin connection $ω_{μ}$ / Christoffel $Γ$
field strength $F_{μν}$	Riemann tensor $R^{ρ}_{σ μν}$
covariant derivative $\partial + A$	covariant derivative $\partial + Γ$
Wilson loop	holonomy / parallel transport around a loop
Yang–Mills action $tr F \land ⋆ F$	Einstein–Hilbert action $\int R - g$

The essential difference: gravity's connection is built from the metric (it acts on the same tangent directions it curves), so the field strength couples to the geometry of spacetime itself, whereas a gauge connection lives on an internal bundle over a fixed spacetime. This is why gravity is not "just another gauge theory", though the geometric language is shared.

Gauss–Bonnet and the Euler class

For a closed surface, integrating the Gaussian curvature gives a topological invariant — the Gauss–Bonnet theorem $\frac{1}{2 π} \int_{M} K d A = χ (M) = 2 - 2 g,$ which is the Euler class (characteristic-classes.md) of the tangent bundle integrated via Chern–Weil. So the same "curvature integral = topological number" principle that gives the instanton number in gauge theory gives the Euler characteristic in geometry. The metric details of this story — geodesics, constant-curvature spaces, the parallel postulate — live in geometry/curvature.md and geometry/constant-curvature.md; this folder supplies the bundle-theoretic frame around them.

References

do Carmo, Riemannian Geometry.
Nakahara, Geometry, Topology and Physics, Ch. 7.
Misner, Thorne & Wheeler, Gravitation — the physics of the tangent-bundle connection.