Curvature in local coordinates bridges abstract geometry with practical calculations. It introduces Christoffel symbols and curvature components, essential tools for understanding how curved spaces behave. These concepts allow us to quantify and analyze the curvature of various geometries.

This section explores constant curvature geometries like spherical and hyperbolic spaces, as well as more complex examples. By examining surfaces of revolution and Lie groups, we see how curvature manifests in diverse mathematical structures, deepening our grasp of curved spaces.

Curvature Components and Christoffel Symbols

Christoffel Symbols and Their Significance

• Christoffel symbols represent connection coefficients in a coordinate basis
• Denoted by $\Gamma^i_{jk}$ where i, j, and k are indices
• Express how basis vectors change as you move along coordinate curves
• Calculate using partial derivatives of the metric tensor $g_{ij}$
• Formula for Christoffel symbols: $\Gamma^i_{jk} = \frac{1}{2}g^{il}(\partial_j g_{kl} + \partial_k g_{jl} - \partial_l g_{jk})$
• Play crucial role in defining covariant derivatives and parallel transport
• Used to compute geodesics, which are paths of shortest distance in curved spaces

Curvature Components in Coordinate Systems

• Riemann curvature tensor expressed in local coordinates as $R^i_{jkl}$
• Components calculated using Christoffel symbols and their derivatives
• Full expression: $R^i_{jkl} = \partial_k \Gamma^i_{jl} - \partial_l \Gamma^i_{jk} + \Gamma^i_{km}\Gamma^m_{jl} - \Gamma^i_{lm}\Gamma^m_{jk}$
• Measures how parallel transport around infinitesimal loops fails to return vectors to their original orientation
• Symmetries of the Riemann tensor reduce independent components
• Bianchi identities further constrain curvature components
• Ricci tensor obtained by contracting Riemann tensor: $R_{ij} = R^k_{ikj}$
• Scalar curvature derived from Ricci tensor: $R = g^{ij}R_{ij}$

Geometries with Constant Curvature

Spherical Geometry and Its Properties

• Positive constant curvature geometry
• Realized on the surface of a sphere with radius R
• Curvature K = 1/R^2, where R represents the sphere's radius
• Great circles serve as geodesics (shortest paths between points)
• Parallel lines eventually intersect
• Sum of angles in a triangle exceeds 180 degrees
• Area of a triangle given by $A = R^2(\alpha + \beta + \gamma - \pi)$, where α, β, γ are angles
• Isometry group SO(3) describes symmetries of the sphere

Hyperbolic Geometry and Its Characteristics

• Negative constant curvature geometry
• Models include Poincaré disk and upper half-plane
• Curvature K = -1/R^2, where R represents the characteristic length scale
• Geodesics appear as circular arcs perpendicular to the boundary (Poincaré disk model)
• Parallel lines diverge
• Sum of angles in a triangle less than 180 degrees
• Area of a triangle given by $A = R^2(\pi - \alpha - \beta - \gamma)$, where α, β, γ are angles
• Isometry group PSL(2,R) describes symmetries of hyperbolic space

Curvature of Symmetric Spaces

• Homogeneous spaces with additional symmetry properties
• Include spheres, hyperbolic spaces, and Euclidean spaces as special cases
• Curvature tensor remains invariant under parallel transport
• Classified into compact and non-compact types
• Compact type (spherical-like) have non-negative sectional curvature
• Non-compact type (hyperbolic-like) have non-positive sectional curvature
• Rank of symmetric space determines number of flat totally geodesic submanifolds
• Cartan classification provides complete list of irreducible symmetric spaces

Curvature of Special Surfaces and Spaces

Curvature of Surfaces of Revolution

• Generated by rotating a curve around an axis
• Metric given in coordinates (u,v): $ds^2 = du^2 + f(u)^2 dv^2$
• Gaussian curvature K and mean curvature H depend on generating curve
• For curve (r(u), z(u)), Gaussian curvature: $K = -\frac{r''}{r(1 + (r')^2)}$
• Mean curvature: $H = \frac{r(1 + (r')^2) + r''z' - r'z''}{2r(1 + (r')^2)^{3/2}}$
• Includes spheres, cylinders, cones, and tori as special cases
• Pseudosphere (tractroid) provides model for hyperbolic geometry

Curvature of Lie Groups and Their Properties

• Lie groups equipped with left-invariant metrics
• Curvature determined by structure constants of the Lie algebra
• Sectional curvature for left-invariant metric: $K(X,Y) = \frac{1}{4}|[X,Y]|^2 - \frac{3}{4}\langle [X,X], [Y,Y] \rangle$
• Bi-invariant metrics on compact Lie groups have non-negative sectional curvature
• Scalar curvature for bi-invariant metric: $R = -\frac{1}{4}\sum_{i,j,k} |c_{ijk}|^2$, where $c_{ijk}$ are structure constants
• Special unitary group SU(2) isometric to 3-sphere with constant positive curvature
• Hyperbolic space can be realized as a quotient of non-compact Lie groups