Curvature

Prerequisites: Manifold. Full path (3 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

Curvature measures how much a space fails to be flat, and it can be detected from inside the space without looking at it from outside.

Imagine ants living on a surface who can only measure lengths and angles along the surface. On a flat sheet, the angles of every triangle add up to 180 degrees. On a sphere they add up to more. Start at the north pole, walk down to the equator, walk a quarter of the way around the equator, and walk back up to the pole. Each corner of this triangle is a right angle, so the angles add up to 270 degrees. The excess of 90 degrees is proportional to the area of the triangle, which is one eighth of the sphere. A triangle twice as large has twice the excess, and on a larger sphere the same excess needs a larger triangle. The excess per unit area, with the angle measured in radians, is the Gaussian curvature, which for a sphere is one over the radius squared.

The same triangle shows a second effect. Carry an arrow around the triangle, always keeping it "as parallel as possible" to its previous direction along the surface (parallel transport). When it returns to the starting point it has turned by 90 degrees. On a flat plane it would come back unchanged. The rotation angle for a small loop is the curvature times the enclosed area.

Curvature in this sense is intrinsic. A sheet of paper rolled into a cylinder looks curved from outside, but the ants measure no angle excess and no rotation, because the cylinder can be unrolled flat without stretching. A sphere cannot, which is why every flat map of the Earth distorts distances somewhere.

In general relativity the curvature of spacetime shows up as tidal effects: two nearby freely falling particles that start out moving parallel accelerate towards or away from each other.

• What Is (Gaussian) Curvature?

Concrete

Examples

The diagram below shows three concrete examples for curvature (Riemann curvature tensor) of simple 2-dimensional manifolds. For a more detailed explanation see Fun with Symmetry.

Abstract

Let $E \to M$ be a vector bundle with a connection $\nabla$ (a covariant derivative). Its curvature is

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

for vector fields $X, Y$. Although built from derivatives, $R^\nabla(X,Y)s$ is $C^\infty(M)$-linear in $X$, $Y$ and in the section $s$, so $R^\nabla$ is a tensor: a 2-form on $M$ with values in $\mathrm{End}(E)$, $F^\nabla \in \Omega^2(M, \mathrm{End}\, E)$. Extending $\nabla$ to the exterior covariant derivative $d_\nabla$ on $E$-valued forms, curvature is the failure of $d_\nabla$ to square to zero:

$$d_\nabla^2\, \sigma = F^\nabla \wedge \sigma .$$

In a local frame where $\nabla = d + A$ with a matrix of 1-forms $A$, $F = dA + A \wedge A$, and under a change of frame $g$, $F \to g^{-1} F g$. The Bianchi identity reads $d_\nabla F = 0$. The connection is flat ($F = 0$) if and only if around every point there is a local frame of parallel sections; parallel transport then depends only on the homotopy class of the path. The same structure on a principal bundle is the curvature $\Omega = d\omega + \frac12 [\omega \wedge \omega]$ of a connection form (Ehresmann connection), and in physics it is the field strength of a gauge field.

Riemannian curvature. For the Levi-Civita connection of a (pseudo-)Riemannian metric, $E = TM$ and the components $R^\rho{}_{\sigma\mu\nu}$, defined by $R(\partial_\mu, \partial_\nu)\partial_\sigma = R^\rho{}_{\sigma\mu\nu}\partial_\rho$, are

$$R^\rho{}_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma} .$$

With the index lowered, $R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu} = -R_{\rho\sigma\nu\mu} = R_{\mu\nu\rho\sigma}$, and $R_{\rho[\sigma\mu\nu]} = 0$. In $n$ dimensions this leaves $n^2(n^2-1)/12$ independent components: 1 for $n = 2$, 6 for $n = 3$, 20 for $n = 4$. Contractions give the Ricci tensor $R_{\sigma\nu} = R^\rho{}_{\sigma\rho\nu}$ and the scalar curvature $R = g^{\sigma\nu} R_{\sigma\nu}$, which enter the Einstein equation. Sign conventions for these contractions vary between textbooks.

For a 2-dimensional Riemannian manifold the only invariant is the Gaussian curvature $K$, with $R_{\rho\sigma\mu\nu} = K (g_{\rho\mu} g_{\sigma\nu} - g_{\rho\nu} g_{\sigma\mu})$ and $R = 2K$. Gauss' Theorema Egregium says that $K$ of a surface in $\mathbb{R}^3$, originally defined as the product of the principal curvatures, depends only on the induced metric. For a closed oriented surface, the Gauss-Bonnet theorem relates it to topology:

$$\int_M K \, dA = 2\pi \chi(M) ,$$

with $\chi$ the Euler characteristic ($\chi = 2$ for the sphere, $0$ for the torus). Its generalization to bundles is Chern-Weil theory, in which invariant polynomials in $F$ give closed forms representing characteristic classes (see cohomology).

Path to this page

1. Calculus
2. Linear Algebra
3. Manifold

Leads to: Einstein Equation, Gauss-Bonnet Theorem.