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
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Christoffel symbols represent connection coefficients in a coordinate basis
Denoted by Γjki 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 gij
Formula for Christoffel symbols: Γjki=21gil(∂jgkl+∂kgjl−∂lgjk)
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 Rjkli
Components calculated using Christoffel symbols and their derivatives
Full expression: Rjkli=∂kΓjli−∂lΓjki+ΓkmiΓjlm−ΓlmiΓjkm
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: Rij=Rikjk
Scalar curvature derived from Ricci tensor: R=gijRij
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=R2(α+β+γ−π), 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=R2(π−α−β−γ), 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): ds2=du2+f(u)2dv2
Gaussian curvature K and mean curvature H depend on generating curve
For curve (r(u), z(u)), Gaussian curvature: K=−r(1+(r′)2)r′′
Mean curvature: H=2r(1+(r′)2)3/2r(1+(r′)2)+r′′z′−r′z′′
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)=41∣[X,Y]∣2−43⟨[X,X],[Y,Y]⟩
Bi-invariant metrics on compact Lie groups have non-negative sectional curvature
Scalar curvature for bi-invariant metric: R=−41∑i,j,k∣cijk∣2, where cijk 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