Curvature

Curvature is the central invariant of Riemannian geometry. It measures, in a precise quantitative sense, how much a manifold deviates from being flat — how much parallel transport around a loop rotates vectors, how much geodesics from a point spread apart or converge, how much the volume of a geodesic ball differs from its Euclidean counterpart. All of these phenomena are governed by a single tensor, the Riemann curvature tensor, which encodes the full local geometry of a Riemannian manifold.

The chapter is organized around increasing coarseness of information. The Riemann tensor is the full local curvature data — it determines everything. Sectional curvature is the curvature of individual tangent 2-planes, a family of scalar invariants parametrized by directions. Ricci curvature is the average of sectional curvatures over all planes containing a given direction, a symmetric 2-tensor controlling volume growth. Scalar curvature is the total average, a single number at each point. Finally, the Gauss-Bonnet theorem extracts a global topological invariant from the integral of curvature — the deepest result of the chapter, connecting local geometry to global topology.

The Riemann Curvature Tensor

Chapter 5 ended with the observation that covariant derivatives do not commute: for vector fields $X, Y, Z$, the quantity $\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$ is tensorial. This is the definition of the Riemann tensor.

[!definition] 6.1 — Riemann Curvature Tensor The Riemann curvature tensor of a Riemannian manifold $(M, g)$ with Levi-Civita connection $\nabla$ is the $(1,3)$-tensor field

$R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$

for $X, Y, Z \in \mathfrak{X}(M)$. In local coordinates $(x^i)$, its components $R^i{}_{jk\ell}$ are defined by

$R\!\left(\frac{\partial}{\partial x^k}, \frac{\partial}{\partial x^\ell}\right)\!\frac{\partial}{\partial x^j} = \sum_i R^i{}_{jk\ell}\, \frac{\partial}{\partial x^i},$

and are given explicitly by

$R^i{}_{jk\ell} = \frac{\partial \Gamma^i_{\ell j}}{\partial x^k} - \frac{\partial \Gamma^i_{k j}}{\partial x^k} + \sum_m \left(\Gamma^i_{km}\Gamma^m_{\ell j} - \Gamma^i_{\ell m}\Gamma^m_{k j}\right).$

The fully covariant form is $R_{ijk\ell} = g_{im} R^m{}_{jk\ell}$.

That $R(X,Y)Z$ is indeed tensorial — that its value at $p$ depends only on $X_p$, $Y_p$, $Z_p$, not on their derivatives — is verified by checking that all derivative terms cancel when one multiplies any argument by a smooth function. This cancellation is precisely the reason for including the $\nabla_{[X,Y]}Z$ correction term.

The Riemann tensor has four slots and satisfies a system of algebraic symmetries that reduce the apparent $n^4$ independent components to $\frac{n^2(n^2-1)}{12}$.

[!theorem] 6.1 — Symmetries of the Riemann Tensor The fully covariant Riemann tensor $R_{ijk\ell} = g(R(\partial_k, \partial_\ell)\partial_j, \partial_i)$ satisfies:

1. Antisymmetry in the last two slots: $R_{ijk\ell} = -R_{ij\ell k}$
2. Antisymmetry in the first two slots: $R_{ijk\ell} = -R_{jik\ell}$
3. Pair symmetry: $R_{ijk\ell} = R_{k\ell ij}$
4. First Bianchi identity: $R_{ijk\ell} + R_{i\ell jk} + R_{ikj\ell} = 0$ (cyclic sum in last three indices is zero — note some texts order slots differently)
5. Second Bianchi identity (differential): $\nabla_m R_{ijk\ell} + \nabla_k R_{ijm\ell} + \nabla_\ell R_{ijkm} = 0$… wait, let me state this correctly: $(\nabla_X R)(Y,Z)W + (\nabla_Y R)(Z,X)W + (\nabla_Z R)(X,Y)W = 0$.

The first Bianchi identity is algebraic and follows from the torsion-free condition on the Levi-Civita connection. The second (differential) Bianchi identity is a differential constraint that implies the contracted Bianchi identity, which in turn forces the divergence of the Einstein tensor to vanish — a key structural fact in general relativity.

In dimension $n$, the independent components number: $1$ (for $n=1$, flat), $1$ (for $n=2$, just the Gaussian curvature), $6$ (for $n=3$), $20$ (for $n=4$, the setting of general relativity).

Sectional Curvature

The Riemann tensor is rich but unwieldy. For a Riemannian manifold, the most natural scalar reduction at each point is the sectional curvature, which measures the curvature of the 2-dimensional surface swept out by geodesics in a tangent plane.

[!definition] 6.2 — Sectional Curvature Let $\sigma \subseteq T_pM$ be a 2-dimensional subspace spanned by linearly independent vectors $u, v$. The sectional curvature of $\sigma$ is

$K(\sigma) = K(u, v) = \frac{g(R(u,v)v,\, u)}{g(u,u)\,g(v,v) - g(u,v)^2}.$

The denominator is the squared area of the parallelogram spanned by $u$ and $v$, making $K(\sigma)$ independent of the choice of basis for $\sigma$.

Sectional curvature is the Gaussian curvature of the surface $\exp_p(\sigma \cap B_\varepsilon(0))$ — the geodesic surface swept in the direction of $\sigma$ — evaluated at $p$. This connects the intrinsic Riemann tensor to the classical notion of curvature for surfaces.

The sign of $K$ controls the qualitative behavior of geodesics near $p$ in the plane $\sigma$:

• $K > 0$ (positive curvature): geodesics in the plane $\sigma$ converge — like meridians on a sphere converging at the poles. The paradigm is $S^n$ with $K \equiv 1/R^2$ (radius $R$).
• $K = 0$ (flat): geodesics in $\sigma$ are locally parallel — no convergence or divergence. The paradigm is $\mathbb{R}^n$ with $K \equiv 0$.
• $K < 0$ (negative curvature): geodesics in $\sigma$ diverge — like lines in the hyperbolic plane. The paradigm is $\mathbb{H}^n$ with $K \equiv -1$.

The Riemann tensor is completely determined by the sectional curvatures of all 2-planes at each point: knowing $K(\sigma)$ for all $\sigma \subseteq T_pM$ uniquely recovers $R_p$. Manifolds where $K$ is the same for all planes at all points are space forms.

[!theorem] 6.2 — Space Forms A simply connected complete Riemannian manifold of constant sectional curvature $\kappa$ is isometric to:

• The sphere $S^n(1/\sqrt{\kappa})$ of radius $1/\sqrt{\kappa}$, for $\kappa > 0$
• Euclidean space $\mathbb{R}^n$, for $\kappa = 0$
• Hyperbolic space $\mathbb{H}^n(\kappa)$, for $\kappa < 0$

These are the three model space forms. Every complete Riemannian manifold of constant sectional curvature $\kappa$ is locally isometric to the corresponding model and is a quotient of it by a discrete group of isometries.

Ricci Curvature

From the Riemann tensor one extracts the Ricci tensor by tracing over one pair of indices.

[!definition] 6.3 — Ricci Tensor and Ricci Curvature The Ricci tensor is the symmetric $(0,2)$-tensor

$\mathrm{Ric}(X, Y) = \mathrm{tr}\bigl[Z \mapsto R(Z, X)Y\bigr] = \sum_i g\bigl(R(e_i, X)Y,\, e_i\bigr),$

where $\{e_i\}$ is any orthonormal basis of $T_pM$. In coordinates, $R_{jk} = \sum_i R^i{}_{jik}$.

The Ricci curvature in a unit direction $u \in T_pM$ is $\mathrm{Ric}(u, u)$. It equals $(n-1)$ times the average of the sectional curvatures $K(u, e_i)$ over all unit vectors $e_i$ orthogonal to $u$:

$\mathrm{Ric}(u, u) = (n-1)\overline{K}(u),$

where $\overline{K}(u)$ is this average sectional curvature.

The Ricci tensor controls the rate at which the volume of a geodesic cone in direction $u$ grows or shrinks relative to flat space. The Bishop-Gromov comparison theorem makes this precise: if $\mathrm{Ric} \geq (n-1)\kappa\, g$ for some constant $\kappa$, then the volume of geodesic balls in $M$ is bounded above by the volume of geodesic balls in the model space of constant sectional curvature $\kappa$. This is one of the main tools in comparison geometry.

A Riemannian manifold is an Einstein manifold if $\mathrm{Ric} = \lambda g$ for some constant $\lambda \in \mathbb{R}$. Spheres, hyperbolic spaces, and flat tori are all Einstein. In general relativity, Einstein’s field equations in vacuum are $\mathrm{Ric} = 0$ (Ricci-flat), and the general equations are $\mathrm{Ric} - \frac{1}{2}Rg = 8\pi G\, T$ where $T$ is the stress-energy tensor.

Scalar Curvature

One further contraction gives the scalar curvature.

[!definition] 6.4 — Scalar Curvature The scalar curvature is the smooth function $\mathrm{scal}: M \to \mathbb{R}$ (also written $R$ or $S$) defined as the trace of the Ricci tensor:

$\mathrm{scal}(p) = \mathrm{tr}_g(\mathrm{Ric}) = \sum_{i} \mathrm{Ric}(e_i, e_i) = \sum_{i \neq j} K(e_i, e_j),$

where $\{e_i\}$ is any orthonormal basis of $T_pM$. It is the sum of all sectional curvatures over pairs of basis vectors, or equivalently $\frac{n(n-1)}{2}$ times the average sectional curvature.

Scalar curvature controls the leading correction to the volume of small geodesic balls:

$\mathrm{Vol}(B_r(p)) = \omega_n r^n \left(1 - \frac{\mathrm{scal}(p)}{6(n+2)} r^2 + O(r^4)\right),$

where $\omega_n$ is the volume of the unit ball in $\mathbb{R}^n$. Positive scalar curvature implies geodesic balls are smaller than their Euclidean counterparts; negative scalar curvature implies they are larger. This is the most direct geometric meaning of scalar curvature.

The contracted second Bianchi identity says $\mathrm{div}(\mathrm{Ric}) = \frac{1}{2}\, d(\mathrm{scal})$, or equivalently $\nabla^j R_{jk} = \frac{1}{2}\nabla_k \mathrm{scal}$. This forces the divergence of the Einstein tensor $G = \mathrm{Ric} - \frac{1}{2}\mathrm{scal}\cdot g$ to vanish: $\mathrm{div}(G) = 0$, which is the mathematical expression of energy-momentum conservation in general relativity.

Jacobi Fields and the Spreading of Geodesics

Curvature controls how nearby geodesics spread apart or converge. This is formalized by Jacobi fields — the infinitesimal variation of a geodesic family.

[!definition] 6.5 — Jacobi Field Let $\gamma: [0, L] \to M$ be a geodesic. A Jacobi field along $\gamma$ is a vector field $J$ along $\gamma$ satisfying the Jacobi equation:

$\frac{D^2 J}{dt^2} + R(J, \dot\gamma)\dot\gamma = 0.$

This is a second-order linear ODE along $\gamma$, so Jacobi fields form a $2n$-dimensional vector space. They arise as the variational fields of smooth families of geodesics: if $\Gamma(s, t)$ is a family of geodesics with $\Gamma(0, t) = \gamma(t)$, then $J(t) = \partial\Gamma/\partial s\big|_{s=0}$ is a Jacobi field.

Jacobi fields measure how geodesics emanating from a point spread or focus. Starting from $p = \gamma(0)$ with $J(0) = 0$ and $DJ/dt|_{t=0} = w$ (a unit vector orthogonal to $\dot\gamma$), the Jacobi field describes the separation between $\gamma$ and the nearby geodesic in direction $w$. For small $t$:

$J(t) = t\, w - \frac{t^3}{6} R(w, \dot\gamma)\dot\gamma + O(t^5).$

The curvature term $R(w, \dot\gamma)\dot\gamma$ — the tidal force — shows that positive sectional curvature $K(w, \dot\gamma) > 0$ causes $J$ to grow more slowly than $t$ (geodesics converge), while negative curvature causes $J$ to grow faster (geodesics diverge).

A point $q = \gamma(t_0)$ is a conjugate point to $p$ along $\gamma$ if there exists a non-zero Jacobi field vanishing at both $p$ and $q$. Conjugate points are where nearby geodesics refocus — the analogue of the focal point of a lens. On $S^n$ of radius $R$, every point has a conjugate point at distance $\pi R$ along any geodesic (the antipodal point). In negative curvature, conjugate points do not exist — geodesics diverge and never refocus.

[!theorem] 6.3 — Bonnet-Myers Theorem If $(M, g)$ is a complete Riemannian manifold with $\mathrm{Ric} \geq (n-1)\kappa\, g$ for some $\kappa > 0$, then:

1. The diameter of $M$ satisfies $\mathrm{diam}(M) \leq \pi/\sqrt{\kappa}$.
2. $M$ is compact with finite fundamental group.

In particular, a complete manifold with strictly positive Ricci curvature is compact.

The proof uses Jacobi fields: the Ricci curvature lower bound forces geodesics to develop conjugate points by time $\pi/\sqrt{\kappa}$, after which they cannot be minimizing. Bonnet-Myers is the Riemannian version of the statement that positive curvature makes the universe finite and closed.

The complementary result: the Cartan-Hadamard theorem says that if $K \leq 0$ everywhere, then the exponential map $\exp_p: T_pM \to M$ is a covering map — there are no conjugate points, and $M$ is diffeomorphic to $\mathbb{R}^n$ if simply connected. Negative curvature forces the manifold to be topologically trivial.

The Gauss-Bonnet Theorem

The Gauss-Bonnet theorem is one of the most profound results in mathematics. It asserts that the integral of curvature over a closed manifold is a topological invariant — a number that does not change under any continuous deformation of the metric.

$\int_M K\, \mathrm{vol}_g = 2\pi\, \chi(M),$

where $K$ is the Gaussian curvature and $\chi(M)$ is the Euler characteristic of $M$.

With boundary. If $M$ has boundary $\partial M$, then

$\int_M K\, \mathrm{vol}_g + \int_{\partial M} \kappa_g\, ds = 2\pi\, \chi(M),$

where $\kappa_g$ is the geodesic curvature of $\partial M$.

The Euler characteristic is a topological invariant: $\chi(S^2) = 2$, $\chi(\mathbb{T}^2) = 0$, $\chi(\Sigma_g) = 2 - 2g$ for a genus-$g$ surface. The theorem says:

• Sphere ($\chi = 2$): $\int K\, dA = 4\pi$ regardless of the metric. Any metric on the sphere must have total positive curvature exactly $4\pi$.
• Torus ($\chi = 0$): $\int K\, dA = 0$. Any metric on the torus has as much positive curvature as negative — they must cancel exactly.
• Higher genus ($\chi < 0$): $\int K\, dA < 0$. Any metric must have net negative curvature.

This forces dramatic consequences. On a sphere, you cannot have $K \leq 0$ everywhere (the integral would be $\leq 0$, contradicting $4\pi > 0$). On a torus, you cannot have $K > 0$ everywhere. The topology of the surface constrains the signs of curvature any metric can have.

Generalization: the Chern-Gauss-Bonnet theorem. For a compact oriented Riemannian manifold of dimension $2n$, the generalization replaces $K\, \mathrm{vol}_g$ with the Pfaffian of the curvature 2-form — a certain polynomial in the curvature tensor — integrated over $M$. In dimension 4:

$\int_M \left(\frac{|\mathrm{Rm}|^2}{8} - \frac{|\mathrm{Ric}|^2}{2} + \frac{\mathrm{scal}^2}{8}\right)\mathrm{vol}_g = 4\pi^2\, \chi(M).$

This is a purely local curvature integral, yet it computes a purely topological quantity. The explanation lies in the theory of characteristic classes: the integrand is a closed differential form representing a cohomology class (the Euler class) that depends only on the topology of $M$, not on the metric.

Curvature as an Obstruction

Curvature is not just a measurement but an obstruction — it prevents certain global structures from existing.

Obstruction to flatness. A Riemannian manifold is locally isometric to $\mathbb{R}^n$ (i.e., flat) if and only if $R \equiv 0$. The vanishing of the Riemann tensor is the integrability condition for the existence of locally parallel frames. This is the Riemannian version of the fact that a vector field $F$ on $\mathbb{R}^3$ is a gradient if and only if $\nabla \times F = 0$.

Obstruction to positive curvature. The Bonnet-Myers theorem shows that positive Ricci curvature forces compactness, ruling out many topological types. The Sphere theorem (Berger-Klingenberg, 1960) says that a simply connected Riemannian manifold with $1/4 < K \leq 1$ is homeomorphic to $S^n$ — pinched positive curvature forces spherical topology.

Obstruction to existence of metrics. The Gauss-Bonnet theorem shows that a surface of genus $g \geq 2$ cannot carry a metric of non-negative curvature everywhere: it must have regions of negative curvature. More dramatically, the Lichnerowicz obstruction (1963) uses the scalar curvature to obstruct the existence of positive scalar curvature metrics: if a compact spin manifold has non-zero $\hat{A}$-genus, it cannot carry a metric with $\mathrm{scal} > 0$.

These obstruction results illustrate the deepest theme in Riemannian geometry: the interplay between local curvature conditions and global topological constraints. Curvature is the bridge between analysis (differential equations for geodesics, Jacobi fields) and topology (Euler characteristic, fundamental groups, cobordism).

Curvature in Coordinates

For computation, the coordinate expressions are indispensable. The Riemann tensor components are:

The Ricci tensor: $R_{jk} = R^i{}_{jik}$ (sum over $i$, with the convention that the first and third indices are contracted).

The scalar curvature: $\mathrm{scal} = g^{jk} R_{jk}$.

In normal coordinates at $p$ (where $g_{ij}(p) = \delta_{ij}$ and $\Gamma^k_{ij}(p) = 0$):

$R_{ijk\ell}(p) = \frac{1}{2}\left(\partial_k \partial_j g_{i\ell} + \partial_\ell \partial_i g_{jk} - \partial_\ell \partial_j g_{ik} - \partial_k \partial_i g_{j\ell}\right)\bigg|_p.$

This shows that the Riemann tensor at $p$ is entirely determined by the second derivatives of the metric at $p$ in normal coordinates — the first derivatives vanish by the normal coordinate condition. Curvature is thus a second-order invariant of the metric, the leading non-trivial local invariant that a Riemannian metric possesses.

With the curvature theory complete, the first two parts of the monograph are finished. Part I built the smooth world — manifolds, tangent bundles, differential forms — the language in which geometry is conducted. Part II equipped it with measurement — the Riemannian metric, the Levi-Civita connection, and curvature. Part III now turns to structure and specialization: the geometry of symmetry (Lie groups), the geometry of general bundles, the geometry of mechanics (symplectic), and the geometry of probability (information geometry).

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The definitive treatment of the Riemann tensor and its symmetries is in Petersen’s Riemannian Geometry (3rd ed., 2016). The Gauss-Bonnet theorem is proved in do Carmo’s Differential Geometry of Curves and Surfaces (1976) for surfaces and in Chern’s original paper (1944) in full generality. Jacobi fields and comparison theorems are developed beautifully in Cheeger and Ebin’s Comparison Theorems in Riemannian Geometry (1975). For curvature obstructions and the Gromov-Lawson theory of positive scalar curvature, see Lawson and Michelsohn’s Spin Geometry (1989).