Differential Geometry

Prerequisites: Manifold. Full path (3 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

Differential geometry is calculus on curved spaces. Ordinary calculus is done in flat space with one global set of coordinates. Many spaces in physics are not like that: the surface of the Earth, spacetime in general relativity, the space of all orientations of a rigid body, the phase space of a mechanical system. Each of them looks like ordinary flat space when we zoom in close enough, but the pieces are glued together in a way that no single coordinate grid can cover. Such a space is called a manifold.

On a manifold we can still talk about velocities, gradients and rates of change, because these only need a small neighbourhood. What we cannot do automatically is compare things at different points. On the Earth, "north" at Berlin and "north" at Sydney point in different directions in space, and there is no unique answer to whether two arrows at different places are parallel. Differential geometry adds the structures that answer such questions: a metric to measure lengths and angles, a connection to say how to carry arrows from one point to another, and curvature to measure how much the answer depends on the path taken.

All four fundamental interactions are described with these tools. In general relativity spacetime is a curved manifold and gravity is its curvature. In gauge theories (electromagnetism and the weak and strong interactions) the gauge field is a connection on an internal space attached to each point, and its curvature is the field strength.

• A pictorial introduction to differential geometry, leading to Maxwell's equations as three pictures by Jonathan Gratus

Concrete

The working tools are coordinates, tensors and a few equations. Points of a manifold are labelled by coordinates $x^\mu$, and all objects transform in a definite way under a change of coordinates (see covariant and contravariant coordinates).

Metric. Lengths are measured with the line element

$$ds^2 = g_{\mu\nu}(x)\, dx^\mu dx^\nu ,$$

and the length of a curve $x^\mu(\lambda)$ is $\int \sqrt{g_{\mu\nu} \dot x^\mu \dot x^\nu}\, d\lambda$.

Connection and geodesics. Derivatives of vector fields are replaced by covariant derivatives, $\nabla_\mu V^\rho = \partial_\mu V^\rho + \Gamma^\rho_{\mu\sigma} V^\sigma$, with the Christoffel symbols of the Levi-Civita connection

$$\Gamma^\rho_{\mu\nu} = \tfrac12 g^{\rho\lambda} \big( \partial_\mu g_{\nu\lambda} + \partial_\nu g_{\mu\lambda} - \partial_\lambda g_{\mu\nu} \big) .$$

The straightest curves, geodesics, satisfy

$$\frac{d^2 x^\rho}{d\lambda^2} + \Gamma^\rho_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda} = 0 .$$

Curvature. The Riemann tensor measures the failure of covariant derivatives to commute, $[\nabla_\mu, \nabla_\nu] V^\rho = R^\rho{}_{\sigma\mu\nu} V^\sigma$ for the Levi-Civita connection.

Worked example: the sphere. For a sphere of radius $R$ in the angles $\theta, \phi$,

$$ds^2 = R^2 \big( d\theta^2 + \sin^2\theta\, d\phi^2 \big) .$$

The only non-zero metric derivative is $\partial_\theta g_{\phi\phi} = 2R^2 \sin\theta \cos\theta$, and the formula above gives

$$\Gamma^\theta_{\phi\phi} = -\sin\theta \cos\theta, \qquad \Gamma^\phi_{\theta\phi} = \Gamma^\phi_{\phi\theta} = \cot\theta ,$$

with all others zero. The geodesic equation for $\theta$ reads $\ddot\theta - \sin\theta\cos\theta\, \dot\phi^2 = 0$. The equator, $\theta = \pi/2$ with $\phi = \lambda$, solves it, while a circle of latitude $\theta = \theta_0 \neq \pi/2$ with $\dot\phi \neq 0$ does not. Working out the Riemann tensor gives $R^\theta{}_{\phi\theta\phi} = \sin^2\theta$, and the Gaussian curvature is $K = 1/R^2$, the same at every point.

In physics these equations appear as the geodesic equation for free fall and the Einstein equation in general relativity, and in a parallel form (covariant derivatives $D_\mu = \partial_\mu + A_\mu$ and field strengths) in gauge theories.

• Differential Geometry in Physics An introductory exposition for true non-experts by Marián Fecko
• Differential Geometry and Lie Groups for Physicists by Marián Fecko
• Manifolds and Differential Forms lecture notes by Reyer Sjamaar

Abstract

A smooth $n$-manifold $M$ is a second-countable Hausdorff space with a maximal atlas of charts to $\mathbb{R}^n$ whose transition maps are $C^\infty$. Smooth functions form the algebra $C^\infty(M)$. Tangent vectors at $p$ can be defined as derivations of germs of functions at $p$, vector fields as derivations of $C^\infty(M)$, and the commutator of two derivations is again one, the Lie bracket $[X, Y]$. The tangent bundle $TM$ and its dual $T^*M$ generate all tensor bundles, and differential forms with the exterior derivative give the de Rham complex, whose cohomology depends only on the topology of $M$.

On top of the smooth structure, differential geometry studies additional structures and their invariants:

• (Pseudo-)Riemannian metrics $g$, a non-degenerate symmetric bilinear form on each tangent space. Each determines a unique torsion-free metric connection, the Levi-Civita connection, and its Riemann curvature tensor (curvature).
• Fiber bundles, in particular principal $G$-bundles and associated vector bundles, with connections (Ehresmann connection), curvature, holonomy and characteristic classes.
• Symplectic structures, closed non-degenerate 2-forms, and Poisson structures.
• Lie groups and their actions on manifolds, which formalize continuous symmetries.

Local invariants (curvature) and global invariants (topology) are related by theorems such as Gauss-Bonnet and its generalization by Chern-Weil theory, which represents characteristic classes of a bundle by polynomials in the curvature of any connection on it.

The dictionary to physics:

• General relativity: spacetime is a 4-dimensional Lorentzian manifold $(M, g)$. The Levi-Civita connection gives free fall as geodesic motion, and the Einstein equation relates the Ricci curvature to the energy-momentum tensor.
• Yang-Mills theory: a gauge field is a connection on a principal $G$-bundle over spacetime, the field strength is its curvature, and gauge transformations are bundle automorphisms covering the identity. Matter fields are sections of associated vector bundles.
• Classical mechanics: the phase space $T^*Q$ of a configuration space $Q$ carries the canonical symplectic form $\sum_i dp_i \wedge dq^i$ (some authors use the opposite sign), and Hamilton's equations are the flow of the vector field $X_H$ defined by $\iota_{X_H} \omega = \pm dH$, with the sign fixed by that convention.

Path to this page

1. Calculus
2. Linear Algebra
3. Manifold

Leads to: Atiyah-Singer Index Theorem.