Riemannian Geometry Unit 3 study guides

Connections and Covariant Derivatives

3.1

Affine connections and their properties

3.2

Levi-Civita connection and its uniqueness

3.3

Covariant derivatives of tensor fields

3.4

Parallel transport along curves

unit 3 review

Connections and covariant derivatives are fundamental concepts in Riemannian geometry, bridging the gap between flat and curved spaces. They provide tools to compare vectors at different points on a manifold, enabling the study of how geometric objects change as they move through curved spaces. These concepts are crucial for understanding geodesics, parallel transport, and curvature. They form the mathematical foundation for describing the geometry of spacetime in general relativity and have applications in various fields of physics and mathematics.

Key Concepts and Definitions

Manifolds are topological spaces that locally resemble Euclidean space and provide a framework for studying curved spaces
Tangent spaces $T_pM$ at each point $p$ on a manifold $M$ consist of tangent vectors that represent infinitesimal displacements
Vector fields assign a tangent vector to each point on the manifold, allowing for the study of smooth functions and differential operators
Connections define a way to transport vectors along curves on the manifold while preserving certain properties (e.g., parallelism, length, or angle)
Covariant derivatives $\nabla_XY$ $\nabla_{X} Y$ measure the change of a vector field $Y$ $Y$ along the direction of another vector field $X$ $X$ , taking into account the curvature of the manifold
- Generalizes the concept of directional derivatives from Euclidean space to curved manifolds
Curvature quantifies the extent to which a manifold deviates from being flat (Euclidean) and is related to the non-commutativity of covariant derivatives
Torsion measures the failure of a connection to be symmetric, i.e., the difference between $\nabla_XY$ and $\nabla_YX$

Connections on Manifolds

Connections provide a way to compare vectors in different tangent spaces and define parallel transport
Affine connections are linear maps $\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)$ $\nabla : Γ (TM) \times Γ (TM) \to Γ (TM)$ satisfying certain properties (linearity, Leibniz rule, and compatibility with the tangent bundle structure)
- $\Gamma(TM)$ denotes the space of smooth vector fields on the manifold $M$
Christoffel symbols $\Gamma^k_{ij}$ $Γ_{ij}^{k}$ are the components of a connection with respect to a chosen coordinate system
- They describe how the basis vectors of the tangent space change as one moves along the manifold
Levi-Civita connection is a unique torsion-free connection that is compatible with the metric (preserves inner products)
- Fundamental in Riemannian geometry and general relativity
Connections can be extended to act on tensor fields of arbitrary rank, allowing for the definition of covariant derivatives on more general objects
Parallel transport along a curve using a connection preserves the direction and magnitude of vectors, but may result in a different vector due to curvature

Parallel Transport and Geodesics

Parallel transport is the process of moving a vector along a curve on a manifold while maintaining its "parallelism" according to a given connection
- Generalizes the notion of parallel lines in Euclidean space to curved manifolds
Geodesics are curves on a manifold that parallel transport their own tangent vectors
- Intuitively, they are the "straightest" possible paths between two points on the manifold
- In Riemannian geometry, geodesics are locally length-minimizing curves
Geodesic equation $\nabla_{\dot{\gamma}}\dot{\gamma} = 0$ $\nabla_{\overset{γ}{˙}} \overset{γ}{˙} = 0$ characterizes geodesics as curves with vanishing covariant acceleration
- $\gamma(t)$ is the parametrized curve and $\dot{\gamma}$ is its tangent vector
Exponential map $\exp_p: T_pM \to M$ $exp_{p} : T_{p} M \to M$ sends a tangent vector $v$ $v$ at point $p$ $p$ to the point reached by following the geodesic starting at $p$ $p$ with initial velocity $v$ $v$ for a unit time
- Provides a local diffeomorphism between the tangent space and the manifold near $p$
Geodesic deviation measures the relative acceleration between nearby geodesics and is related to the curvature of the manifold
Parallel transport and geodesics play a crucial role in understanding the geometry of curved spaces and have applications in physics (e.g., general relativity)

Covariant Derivatives

Covariant derivatives extend the concept of directional derivatives to vector fields on manifolds, taking into account the curvature of the space
For vector fields $X$ $X$ and $Y$ $Y$ , the covariant derivative $\nabla_XY$ $\nabla_{X} Y$ measures the change of $Y$ $Y$ along the flow of $X$ $X$
- Generalizes the Euclidean directional derivative $X(Y) = \sum_i X^i \frac{\partial Y}{\partial x^i}$
In local coordinates, the covariant derivative is expressed using Christoffel symbols: $\nabla_XY = \sum_i X^i \left(\frac{\partial Y^k}{\partial x^i} + \sum_{j} \Gamma^k_{ij} Y^j\right) \frac{\partial}{\partial x^k}$
Covariant derivatives satisfy several properties:
- Linearity: $\nabla_{aX+bY}Z = a\nabla_XZ + b\nabla_YZ$ and $\nabla_X(aY+bZ) = a\nabla_XY + b\nabla_XZ$
- Leibniz rule (product rule): $\nabla_X(fY) = (Xf)Y + f\nabla_XY$ for a smooth function $f$
- Compatibility with the metric: $X\langle Y, Z\rangle = \langle \nabla_XY, Z\rangle + \langle Y, \nabla_XZ\rangle$ (for Riemannian manifolds)
Covariant derivatives can be extended to act on tensor fields of arbitrary rank using the Leibniz rule and linearity
- Allows for the study of differential equations and geometric properties of tensor fields on manifolds
The covariant derivative of the metric tensor $g$ $g$ vanishes identically for the Levi-Civita connection: $\nabla g = 0$ $\nabla g = 0$
- Ensures that parallel transport preserves inner products and lengths of vectors

Curvature and Torsion

Curvature measures the non-commutativity of covariant derivatives and the deviation of a manifold from being flat (Euclidean)
Riemann curvature tensor $R(X, Y)Z = \nabla_X\nabla_YZ - \nabla_Y\nabla_XZ - \nabla_{[X, Y]}Z$ $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ quantifies the curvature of a Riemannian manifold
- $[X, Y] = XY - YX$ is the Lie bracket of vector fields $X$ and $Y$
- Measures the difference between parallel transporting a vector along two different paths
In local coordinates, the components of the Riemann curvature tensor are given by $R^l_{ijk} = \frac{\partial \Gamma^l_{jk}}{\partial x^i} - \frac{\partial \Gamma^l_{ik}}{\partial x^j} + \sum_m (\Gamma^l_{im}\Gamma^m_{jk} - \Gamma^l_{jm}\Gamma^m_{ik})$
Ricci curvature $\operatorname{Ric}(X, Y) = \operatorname{tr}(Z \mapsto R(X, Z)Y)$ $Ric (X, Y) = tr (Z \mapsto R (X, Z) Y)$ is a contraction of the Riemann curvature tensor
- Measures the average curvature in different directions at a point
Scalar curvature $S = \operatorname{tr}(\operatorname{Ric})$ $S = tr (Ric)$ is the trace of the Ricci curvature tensor
- Provides a single scalar value that characterizes the curvature at each point
Torsion tensor $T(X, Y) = \nabla_XY - \nabla_YX - [X, Y]$ $T (X, Y) = \nabla_{X} Y - \nabla_{Y} X - [X, Y]$ measures the failure of a connection to be symmetric
- Vanishes identically for the Levi-Civita connection
Curvature and torsion provide important geometric insights and have applications in various areas of mathematics and physics
- Einstein field equations in general relativity relate the curvature of spacetime to the presence of matter and energy

Applications in Physics

General relativity describes gravity as the curvature of spacetime, with the metric tensor encoding the geometry
- Geodesics in spacetime correspond to the paths of freely falling particles and light rays
- Einstein field equations $G_{\mu\nu} = 8\pi T_{\mu\nu}$ relate the Einstein tensor $G_{\mu\nu}$ (constructed from the Ricci curvature and scalar curvature) to the stress-energy tensor $T_{\mu\nu}$ (describing matter and energy)
Parallel transport and covariant derivatives are essential for formulating physical laws in curved spacetime
- Conservation laws and equations of motion must be expressed using covariant derivatives to ensure their validity in arbitrary coordinate systems
Geodesic deviation equation $\frac{D^2\xi^{\mu}}{d\tau^2} = -R^{\mu}_{\nu\rho\sigma}u^{\nu}u^{\rho}\xi^{\sigma}$ $\frac{D ^{2} ξ ^{μ}}{d τ ^{2}} = - R_{ν ρ σ}^{μ} u^{ν} u^{ρ} ξ^{σ}$ describes the relative acceleration of nearby geodesics due to spacetime curvature
- $\xi^{\mu}$ is the deviation vector, $u^{\mu}$ is the tangent vector to the geodesic, and $\frac{D}{d\tau}$ denotes the covariant derivative along the geodesic
- Explains tidal forces and the stretching and squeezing of matter in gravitational fields
Curvature singularities, such as those found in black holes and at the Big Bang, are regions where the spacetime curvature becomes infinite
- Indicate the breakdown of classical general relativity and the need for a quantum theory of gravity
Connections and covariant derivatives also play a role in gauge theories, such as electromagnetism and the Standard Model of particle physics
- Gauge fields (e.g., the electromagnetic potential) can be interpreted as connections on principal bundles, with the curvature corresponding to the field strength (e.g., the electromagnetic field tensor)

Examples and Calculations

Sphere $S^2$ $S^{2}$ with the induced metric from $\mathbb{R}^3$ $R^{3}$ :
- Christoffel symbols: $\Gamma^{\theta}_{\phi\phi} = -\sin\theta\cos\theta$ , $\Gamma^{\phi}_{\theta\phi} = \Gamma^{\phi}_{\phi\theta} = \cot\theta$ , others zero
- Geodesics: great circles (e.g., equator, meridians)
- Riemann curvature tensor: $R_{\theta\phi\theta\phi} = \sin^2\theta$ , others zero or related by symmetries
- Scalar curvature: $S = \frac{2}{r^2}$ , where $r$ is the radius of the sphere
Hyperbolic plane $\mathbb{H}^2$ $H^{2}$ with the Poincaré metric $ds^2 = \frac{dx^2 + dy^2}{y^2}$ $d s^{2} = \frac{d x ^{2} + d y ^{2}}{y ^{2}}$ :
- Christoffel symbols: $\Gamma^x_{yy} = -\frac{1}{y}$ , $\Gamma^y_{xy} = \Gamma^y_{yx} = \frac{1}{y}$ , others zero
- Geodesics: vertical lines and semicircles orthogonal to the boundary at infinity
- Riemann curvature tensor: $R_{xyxy} = -\frac{1}{y^2}$ , others zero or related by symmetries
- Scalar curvature: $S = -2$
Schwarzschild spacetime describing a non-rotating, uncharged black hole:
- Metric: $ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1}dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$
- Christoffel symbols: $\Gamma^t_{tr} = \frac{M}{r^2}\left(1-\frac{2M}{r}\right)^{-1}$ , $\Gamma^r_{tt} = \frac{M}{r^2}\left(1-\frac{2M}{r}\right)$ , $\Gamma^r_{rr} = -\frac{M}{r^2}\left(1-\frac{2M}{r}\right)^{-1}$ , $\Gamma^r_{\theta\theta} = -r\left(1-\frac{2M}{r}\right)$ , $\Gamma^r_{\phi\phi} = -r\sin^2\theta\left(1-\frac{2M}{r}\right)$ , $\Gamma^{\theta}_{r\theta} = \Gamma^{\theta}_{\theta r} = \frac{1}{r}$ , $\Gamma^{\theta}_{\phi\phi} = -\sin\theta\cos\theta$ , $\Gamma^{\phi}_{r\phi} = \Gamma^{\phi}_{\phi r} = \frac{1}{r}$ , $\Gamma^{\phi}_{\theta\phi} = \Gamma^{\phi}_{\phi\theta} = \cot\theta$
- Geodesics: orbits of test particles and light rays in the gravitational field of the black hole
- Ricci curvature: $R_{tt} = -\frac{2M}{r^3}$ , $R_{rr} = \frac{2M}{r^3}\left(1-\frac{2M}{r}\right)^{-1}$ , $R_{\theta\theta} = 2M$ , $R_{\phi\phi} = 2M\sin^2\theta$
- Scalar curvature: $S = 0$ (vacuum solution)