5.3 Sectional, Ricci, and scalar curvatures

Curvatures in Riemannian geometry measure how a manifold deviates from being flat. Sectional curvature measures the Gaussian curvature of geodesic planes, Ricci curvature averages sectional curvatures, and scalar curvature provides a single value for total curvature at each point.

These curvatures are interconnected and offer complementary information about a manifold's geometry. They have intuitive interpretations: sectional curvature controls geodesic spreading, Ricci curvature affects volume growth, and scalar curvature relates to large-scale volume behavior and general relativity.

Definitions of curvatures

• Curvatures are fundamental invariants in Riemannian geometry that measure the deviation of a manifold from being flat
• They provide important information about the intrinsic geometry and shape of a Riemannian manifold

Sectional curvature

• Measures the Gaussian curvature of geodesic planes in a Riemannian manifold
• Determined by the Riemannian curvature tensor $R(X,Y,Z,W)$ evaluated on pairs of orthonormal vectors $X,Y$
• Formula: $K(X,Y) = \frac{R(X,Y,Y,X)}{|X \wedge Y|^2}$
• Provides the most complete description of curvature at a point

Ricci curvature

• Obtained by taking the trace of the Riemannian curvature tensor
• Measures the average sectional curvature in each tangent direction
• Formula: $Ric(X,Y) = \sum_{i=1}^n R(X,e_i,e_i,Y)$ for an orthonormal basis $\{e_i\}$
• Captures the amount of volume distortion in the manifold

Scalar curvature

• Further contraction of the Ricci curvature
• Defined as the trace of the Ricci tensor
• Formula: $S = \sum_{i=1}^n Ric(e_i,e_i)$ for an orthonormal basis $\{e_i\}$
• Provides a single real number that represents the total curvature at each point

Relationships between curvatures

• The different notions of curvature are interconnected and provide complementary information about the geometry of a Riemannian manifold

Ricci curvature vs sectional curvature

• Ricci curvature is the average of sectional curvatures over all planes containing a given direction
• If the sectional curvature is constant (space forms), then the Ricci curvature is proportional to the metric tensor
• Positive (negative) Ricci curvature implies the existence of some positive (negative) sectional curvatures

Scalar curvature vs Ricci curvature

• Scalar curvature is the trace of the Ricci tensor
• Positive (negative) scalar curvature implies the existence of some positive (negative) Ricci curvatures
• In dimension 2, the scalar curvature is twice the Gaussian curvature
• The scalar curvature appears in the Hilbert-Einstein functional in general relativity

Geometric interpretations

• The different curvatures have intuitive geometric interpretations that help understand their meaning

Sectional curvature interpretation

• Measures the Gaussian curvature of geodesic planes
• Controls the spreading or focusing of geodesics and the growth of Jacobi fields
• Positive (negative) sectional curvature implies that geodesics tend to converge (diverge)

Ricci curvature interpretation

• Measures the average convergence or divergence of geodesics in each direction
• Controls the volume growth of small balls in the manifold
• Positive (negative) Ricci curvature implies that the volume of balls grows slower (faster) than in Euclidean space

Scalar curvature interpretation

• Provides a rough measure of the total curvature at each point
• Related to the asymptotic behavior of the volume of large balls
• Appears in the Hilbert-Einstein functional, which is the action functional for general relativity

Computations and formulas

• Computing the different curvatures involves the Riemannian curvature tensor and its contractions

Sectional curvature formula

• $K(X,Y) = \frac{R(X,Y,Y,X)}{|X \wedge Y|^2}$ for orthonormal vectors $X,Y$
• Can be computed using the Christoffel symbols and their derivatives
• In local coordinates: $K(\partial_i,\partial_j) = R_{ijij}$

Ricci curvature formula

• $Ric(X,Y) = \sum_{i=1}^n R(X,e_i,e_i,Y)$ for an orthonormal basis $\{e_i\}$
• In local coordinates: $Ric_{ij} = \sum_{k=1}^n R_{ikjk}$
• Can be expressed in terms of the Christoffel symbols and the metric tensor

Scalar curvature formula

• $S = \sum_{i=1}^n Ric(e_i,e_i)$ for an orthonormal basis $\{e_i\}$
• In local coordinates: $S = g^{ij}Ric_{ij}$
• Can be computed using the metric tensor and its derivatives

Curvature in special cases

• Certain classes of Riemannian manifolds have special curvature properties that simplify their geometry

Curvature of surfaces

• For 2-dimensional Riemannian manifolds (surfaces), the sectional curvature and Gaussian curvature coincide
• The scalar curvature is twice the Gaussian curvature
• Gauss's Theorema Egregium states that the Gaussian curvature is an intrinsic invariant

Curvature of space forms

• Space forms are Riemannian manifolds with constant sectional curvature (spheres, Euclidean spaces, hyperbolic spaces)
• The Ricci curvature is proportional to the metric tensor: $Ric = (n-1)Kg$
• The scalar curvature is a constant multiple of the sectional curvature: $S = n(n-1)K$

Curvature of product manifolds

• The sectional curvature of a product manifold $(M_1 \times M_2, g_1 \oplus g_2)$ is determined by the curvatures of the factors
• The Ricci curvature of a product manifold is the direct sum of the Ricci curvatures of the factors
• The scalar curvature of a product manifold is the sum of the scalar curvatures of the factors

Curvature and topology

• The curvature of a Riemannian manifold is closely related to its topology and can provide obstructions to the existence of certain geometric structures

Influence on Betti numbers

• The Betti numbers $b_k(M)$ measure the rank of the homology groups $H_k(M;\mathbb{R})$
• The Bochner technique relates the Ricci curvature to the vanishing of Betti numbers
• Positive Ricci curvature implies the vanishing of Betti numbers in certain degrees

Influence on fundamental group

• The fundamental group $\pi_1(M)$ captures the non-trivial loops in a manifold
• The Cartan-Hadamard theorem states that a complete, simply connected manifold with non-positive sectional curvature is diffeomorphic to $\mathbb{R}^n$
• Negative Ricci curvature implies the fundamental group is infinite

Gauss-Bonnet theorem

• Relates the total integral of the Gaussian curvature to the Euler characteristic $\chi(M)$ for compact surfaces
• Formula: $\int_M K dA = 2\pi \chi(M)$
• Generalizes to higher dimensions using the Pfaffian of the curvature form

Variational properties

• Curvatures arise naturally in variational problems on the space of Riemannian metrics

Hilbert-Einstein functional

• The Hilbert-Einstein functional is the action functional for general relativity
• Defined as $\mathcal{S}(g) = \int_M (S_g + L_m) dV_g$, where $L_m$ is the Lagrangian density of matter
• Critical points of the functional are Einstein metrics, which satisfy $Ric_g = \lambda g$ for some constant $\lambda$

Yamabe problem

• Aims to find a metric of constant scalar curvature in a given conformal class
• Equivalent to solving the nonlinear PDE $-\Delta_g u + \frac{n-2}{4(n-1)}S_g u = \lambda u^{\frac{n+2}{n-2}}$
• Solved affirmatively by Yamabe, Trudinger, Aubin, and Schoen

Ricci flow

• A geometric evolution equation that deforms a Riemannian metric in the direction of its Ricci curvature
• Defined as $\frac{\partial}{\partial t}g(t) = -2Ric_{g(t)}$
• Used by Perelman to prove the Poincaré conjecture and Thurston's geometrization conjecture

Applications and examples

• Curvature plays a central role in various areas of mathematics and physics

General relativity

• In general relativity, spacetime is modeled as a 4-dimensional Lorentzian manifold
• The Einstein field equations relate the Ricci curvature to the energy-momentum tensor of matter
• Solutions to the Einstein equations describe the geometry of spacetime in the presence of matter and energy

Comparison geometry

• Comparison theorems relate the curvature of a manifold to the geometry of model spaces (space forms)
• The Rauch comparison theorem compares Jacobi fields on a manifold to those on a space form
• The Bishop-Gromov volume comparison theorem compares the volume growth of balls to that in a space form

Sphere theorems

• Sphere theorems provide sufficient conditions for a manifold to be homeomorphic or diffeomorphic to a sphere
• The 1/4-pinched sphere theorem states that a complete, simply connected manifold with $\frac{1}{4} < K \leq 1$ is homeomorphic to a sphere
• The positive mass theorem in general relativity is a type of sphere theorem in Lorentzian geometry