Riemannian geometry — metric, connection, and curvature

Intuition Beginner

A Riemannian manifold is a smooth space equipped with a way to measure lengths and angles, the way a flat piece of paper lets you measure distances with a ruler. Each point carries its own infinitesimal ruler, and the rulers vary smoothly as you move around.

The connection tells you how to compare rulers at nearby points, so you can roll a vector from one place to another without twisting it arbitrarily. Curvature measures whether that rolling path is path-dependent: on a sphere, a vector carried around a triangle comes back rotated, and the size of that rotation is the curvature.

Riemannian geometry is the setting for general relativity, the geometry of surfaces, and the index theorems that connect shape to topology.

Visual Beginner

A curved surface with tangent planes attached at each point, and an arrow being carried around a loop.

The rotation of the arrow after a round trip is the integrated curvature enclosed by the loop.

Worked example Beginner

On a sphere of radius $R$, parallel-transport an arrow around a triangle with three right angles (an octant). The arrow comes back rotated by a right angle.

Step 1. Walk from the north pole to the equator along a meridian, keeping the arrow tangent.

Step 2. Turn along the equator through a quarter of the globe; the arrow stays tangent to the surface.

Step 3. Return up a second meridian to the north pole. The arrow now points a quarter-turn away from where it started.

What this tells us: curvature is detected by parallel transport around closed loops, and the rotation angle is the area enclosed divided by $R\times R$.

Check your understanding Beginner

Formal definition Intermediate+

A Riemannian metric $g$ on a smooth manifold $M$ is a smooth choice of positive-definite inner product $g_p$ on each tangent space $T_{p}M$. The pair $(M,g)$ is a Riemannian manifold ^{[Lee Ch. 2]}.

An affine connection $\nabla$ on $TM$ assigns to each vector field $X$ an operator ${\nabla }_X$ on vector fields, satisfying linearity and the Leibniz rule ${\nabla }_X(fY)=(Xf)Y+f{\nabla }_{X}Y$. The torsion is $T(X,Y)={\nabla }_{X}Y-{\nabla }_{Y}X-[X,Y]$; the connection is torsion-free when $T=0$. The connection is metric-compatible when $\nabla g=0$.

The Riemann curvature tensor is $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$. The sectional curvature of a 2-plane $\sigma \subset T_{p}M$ is $K(\sigma )=⟨R(u,v)v,u⟩/(\Vert u{\Vert }^2\Vert v{\Vert }^2-⟨u,v{⟩}^2)$ for any basis $u,v$ of $\sigma$.

Counterexamples to common slips

• A metric is not a distance function. The Riemannian metric is an inner-product field; the induced geodesic distance is a derived notion.
• Connections are not unique. Torsion-free plus metric-compatibility is the condition that selects the Levi-Civita connection; without those, infinitely many connections exist.
• Curvature is a tensor, not a number. Only after evaluating $R$ on triples of vectors does one obtain numbers like sectional curvature.

Key theorem with proof Intermediate+

Theorem (Fundamental theorem of Riemannian geometry). On every Riemannian manifold $(M,g)$ there is a unique torsion-free, metric-compatible connection $\nabla$, the Levi-Civita connection. It is characterised by the Koszul formula

$$2⟨{\nabla }_{X}Y,Z⟩=X⟨Y,Z⟩+Y⟨Z,X⟩-Z⟨X,Y⟩-⟨X,[Y,Z]⟩+⟨Y,[Z,X]⟩+⟨Z,[X,Y]⟩.$$

Proof. Suppose $\nabla$ is torsion-free and metric-compatible. Expanding $X⟨Y,Z⟩=⟨{\nabla }_{X}Y,Z⟩+⟨Y,{\nabla }_{X}Z⟩$ cyclically in $X,Y,Z$ and using ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]$ (torsion-free) gives the Koszul formula after adding and subtracting the three expansions. The formula determines ${\nabla }_{X}Y$ because $g$ is non-degenerate, proving uniqueness. Running the construction backwards defines an operator satisfying the connection axioms, torsion-freeness, and metric compatibility, proving existence. $\square$

Bridge. This uniqueness result builds toward 03.09.08 (the Dirac operator), where the Levi-Civita connection lifts to spinors and produces the geometric operator whose index is the subject of 03.09.10 (Atiyah-Singer), and appears again in 03.07.05 (Yang-Mills), where the curvature of a connection on a principal bundle is the field strength that the action integrates. The foundational reason a canonical connection matters is that geometric invariants of $(M,g)$ — curvature, geodesics, the Laplacian — must be intrinsic to the metric, and the Levi-Civita connection is exactly the device that makes them so; putting these together, this is the bridge from the smooth-manifold substrate to all the geometric operators and characteristic classes that follow, and the pattern generalises to connections on arbitrary principal bundles in gauge theory.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none reflects that Mathlib's Riemannian infrastructure, while extensive, has not yet been integrated into a curriculum-level formalisation of the comparison theorems and index-theoretic applications downstream units require. Statement-level formalisation of the Levi-Civita uniqueness theorem is within reach.

Advanced results Master

The fundamental theorem selects a canonical connection, but the deep results of Riemannian geometry concern how curvature controls global topology. The Cartan-Hadamard theorem states that a complete manifold of nonpositive sectional curvature has a universal cover diffeomorphic to ${\mathbb{R}}^n$ via the exponential map. The Bonnet-Myers theorem bounds the diameter of a complete manifold with sectional curvature bounded below by a positive constant, forcing finite fundamental group. The Bishop-Gromov volume comparison bounds the volume growth of geodesic balls in terms of a model of constant curvature, leading to rigidity phenomena ^{[Petersen Ch. 9]}.

The Hodge theorem identifies the space of harmonic $k$-forms on a compact Riemannian manifold with the $k$-th de Rham cohomology, realising topological invariants as solutions of an elliptic equation — the bridge to the analytic index theory of 03.09.10. Bochner technique uses the curvature term in the Weitzenböck formula to vanish cohomology under curvature hypotheses.

Synthesis. Riemannian geometry is the load-bearing geometric setting for the analytic half of the curriculum: the Levi-Civita connection established here builds toward 03.09.08 where it lifts to spinors and defines the Dirac operator, the curvature tensor appears again in 03.06.04 as the local input to characteristic classes via Chern-Weil, the Hodge theorem is exactly the bridge that identifies harmonic forms with de Rham cohomology and motivates the index theorem 03.09.10, the foundational reason a canonical connection matters is that intrinsic invariants require a canonical parallel transport, and the bridge is that the same metric-compatible-torsion-free pattern recurs in gauge theory 03.07.05 where principal-bundle connections generalise the Levi-Civita connection; putting these together, Riemannian geometry is the substrate on which index theory, gauge theory, and Hodge theory all stand, and the pattern generalises to pseudo-Riemannian metrics in general relativity 13.03.01.

Full proof set Master

Proposition (Geodesics locally minimise length). For any point $p$ in a Riemannian manifold, there is $\epsilon >0$ such that the exponential map ${\exp }_p$ is a diffeomorphism on the ball of radius $\epsilon$ in $T_{p}M$, and the radial geodesics from $p$ minimise length between their endpoints.

Proof sketch. The exponential map has invertible differential at $0\in T_{p}M$ (its derivative there is the identity via the identification $T_0(T_{p}M)\cong T_{p}M$), so by the inverse function theorem it is a diffeomorphism on some ball. By the Gauss lemma (Exercise 8), radial geodesics are orthogonal to geodesic spheres, so the length of any curve between $p$ and a nearby point $q$ is bounded below by the radial geodesic length, with equality only along the radial geodesic. $\square$

Connections Master

• Dirac operator and index theory 03.09.08, 03.09.10. The Levi-Civita connection lifts to spinor bundles and yields the Dirac operator whose analytic index — the Atiyah-Singer theorem — is computed by integrating curvature characteristic classes.

• Characteristic classes via Chern-Weil 03.06.04, 03.06.06. The curvature 2-form of the Levi-Civita connection feeds invariant polynomials producing Pontryagin and Euler classes; this is the local-to-global bridge from Riemannian curvature to topological invariants.

• Hodge theory 04.09.01 (and Voisin's complex-geometric form). On a compact Kähler manifold, the Hodge decomposition identifies harmonic forms with cohomology, the deep application of the elliptic theory on which Riemannian geometry floats.

• General relativity 13.03.01. Pseudo-Riemannian metrics of Lorentzian signature replace the positive-definite metric; the Einstein equations identify curvature with the stress-energy of matter.

Historical & philosophical context Master

Bernhard Riemann's 1854 Habilitationsvortrag Über die Hypothesen, welche der Geometrie zu Grunde liegen introduced the idea of a manifold carrying a variable inner product, generalising Gauss's intrinsic surface geometry to arbitrary dimension ^{[Riemann 1854]}. The conceptual leap was to make the metric a local, smoothly varying structure rather than a fixed ambient datum — the founding move of intrinsic differential geometry.

The fundamental theorem (uniqueness of the Levi-Civita connection) was articulated by Levi-Civita in 1917 for surfaces and generalised by Weyl in 1918; it crystallised the principle that geometric invariants must be intrinsic. The comparison theorems of the mid-twentieth century (Cartan-Hadamard, Bonnet-Myers, Bishop-Gromov) established that curvature controls global topology — the central insight driving modern geometric analysis and the index-theoretic applications that follow.

Bibliography Master

@book{Lee1997,
  author = {Lee, Jeffrey M.},
  title = {Riemannian Manifolds: An Introduction to Curvature},
  publisher = {Springer GTM 176},
  year = {1997},
}

@book{doCarmo1992,
  author = {do Carmo, Manfredo},
  title = {Riemannian Geometry},
  publisher = {Birkhäuser},
  year = {1992},
}

@book{Petersen2016,
  author = {Petersen, Peter},
  title = {Riemannian Geometry},
  edition = {3},
  publisher = {Springer GTM 171},
  year = {2016},
}

@incollection{Riemann1854,
  author = {Riemann, Bernhard},
  title = {Über die Hypothesen, welche der Geometrie zu Grunde liegen},
  year = {1854},
}