🌀Riemannian Geometry Unit 2 – Riemannian Metrics & Induced Distances

Key Concepts and Definitions

• Riemannian manifold consists of a smooth manifold $M$ equipped with a Riemannian metric $g$
• Riemannian metric $g$ assigns an inner product $g_p$ to each tangent space $T_pM$ at every point $p \in M$
  • Inner product varies smoothly as the point $p$ varies
• Tangent space $T_pM$ vector space attached to each point $p$ of the manifold $M$
  • Tangent vectors represent velocities of curves passing through the point $p$
• Smooth manifold topological space that locally resembles Euclidean space near each point
  • Transition maps between overlapping coordinate charts are smooth functions
• Induced distance function $d(p,q)$ measures the shortest path length between points $p$ and $q$ on the manifold
• Geodesic shortest path between two points on the manifold, generalizing the concept of a straight line in Euclidean space
• Curvature measures the deviation of the manifold from being flat (Euclidean)
  • Quantifies how parallel transport of vectors along closed loops fails to preserve their angle

Riemannian Manifolds Basics

• Riemannian manifold $(M, g)$ fundamental object of study in Riemannian geometry
  • Manifold $M$ represents the underlying space
  • Riemannian metric $g$ provides geometric structure
• Dimension $n$ of a Riemannian manifold number of coordinates needed to specify a point locally
  • Examples: surface ($n=2$), space ($n=3$), spacetime ($n=4$)
• Coordinate chart $(U, \varphi)$ consists of an open subset $U \subset M$ and a homeomorphism $\varphi: U \to \mathbb{R}^n$
  • Allows expressing points in $U$ using $n$ real coordinates
• Atlas collection of coordinate charts that cover the entire manifold $M$
  • Transition maps between overlapping charts must be smooth
• Tangent bundle $TM$ disjoint union of all tangent spaces $T_pM$ of the manifold $M$
• Riemannian manifolds equipped with a natural volume form $dV_g$ induced by the Riemannian metric $g$
  • Enables integration of functions over the manifold

Riemannian Metrics: Properties and Examples

• Riemannian metric $g$ assigns an inner product $\langle \cdot, \cdot \rangle_p$ to each tangent space $T_pM$
  • Inner product symmetric, bilinear, and positive-definite
• Components of the Riemannian metric $g_{ij}$ in local coordinates express the inner product of basis vectors $\partial_i$ and $\partial_j$
  • Matrix representation $[g_{ij}]$ symmetric and positive-definite at each point
• Euclidean metric $g_E$ on $\mathbb{R}^n$ assigns the standard dot product to each tangent space
  • Components $g_{ij} = \delta_{ij}$ (Kronecker delta)
• Induced metric on a submanifold $S \subset M$ obtained by restricting the ambient Riemannian metric to tangent spaces of $S$
  • Example: sphere $S^2$ with induced metric from Euclidean $\mathbb{R}^3$
• Conformal change of metric multiplying a Riemannian metric $g$ by a positive smooth function $f$
  • Preserves angles between vectors but changes lengths
• Product metric on a product manifold $M \times N$ obtained by summing the metrics on $M$ and $N$
  • Example: torus $S^1 \times S^1$ with product metric from circle $S^1$
• Riemannian submersion $\pi: (M, g_M) \to (N, g_N)$ maps Riemannian manifolds such that the differential $d\pi$ is an isometry on horizontal subspaces
  • Useful for constructing new Riemannian manifolds with symmetries

Induced Distances and Length Spaces

• Riemannian metric $g$ induces a distance function $d(p,q)$ on the manifold $M$
  • Measures the infimum of lengths of piecewise smooth curves connecting $p$ and $q$
• Length of a piecewise smooth curve $\gamma: [a,b] \to M$ given by the integral of the norm of its velocity vector $\gamma'(t)$
  • $L(\gamma) = \int_a^b \sqrt{g_{\gamma(t)}(\gamma'(t), \gamma'(t))} dt$
• Riemannian manifold $(M, g)$ becomes a metric space $(M, d)$ with the induced distance function $d$
  • Satisfies axioms of a metric: non-negativity, symmetry, and triangle inequality
• Hopf-Rinow theorem states that a connected Riemannian manifold is complete as a metric space if and only if it is geodesically complete
  • Completeness ensures the existence of minimal geodesics between any two points
• Length space $(X, d)$ generalizes the concept of a Riemannian manifold
  • Distance function $d$ defined as the infimum of lengths of admissible paths
• Intrinsic metric on a length space induced by the length structure
  • Makes the space a metric space and generalizes the Riemannian distance
• Alexandrov space length space with curvature bounded below in the sense of comparison triangles
  • Generalizes Riemannian manifolds with lower curvature bounds

Geodesics and Shortest Paths

• Geodesic $\gamma: I \to M$ locally length-minimizing curve on a Riemannian manifold $(M, g)$
  • Satisfies the geodesic equation $\nabla_{\gamma'}\gamma' = 0$, where $\nabla$ is the Levi-Civita connection
• Exponential map $\exp_p: T_pM \to M$ maps tangent vectors at $p$ to points on the manifold along geodesics
  • Geodesics through $p$ correspond to straight lines in $T_pM$ under the exponential map
• Minimizing geodesic shortest path between two points on the manifold
  • May not be unique, as in the case of antipodal points on a sphere
• Cut locus $C_p$ of a point $p$ set of points where geodesics from $p$ cease to be minimizing
  • Exponential map is a diffeomorphism from a star-shaped domain in $T_pM$ to $M \setminus C_p$
• Geodesic triangle formed by three minimizing geodesics connecting three points on the manifold
  • Angles and side lengths satisfy triangle comparison theorems based on curvature bounds
• Jacobi fields measure the deviation of nearby geodesics
  • Satisfy the Jacobi equation, a second-order linear ODE involving the Riemann curvature tensor
• Rauch comparison theorem relates the growth of Jacobi fields to the curvature of the manifold
  • Used to prove the Toponogov triangle comparison theorem and the sphere theorem

Curvature in Riemannian Geometry

• Riemann curvature tensor $R(X, Y)Z$ measures the non-commutativity of the covariant derivative $\nabla$
  • Quantifies the failure of parallel transport along infinitesimal parallelograms
• Sectional curvature $K(X, Y)$ measures the Gaussian curvature of the 2-plane spanned by orthonormal vectors $X$ and $Y$
  • Computed as $K(X, Y) = \frac{\langle R(X, Y)Y, X \rangle}{\langle X, X \rangle \langle Y, Y \rangle - \langle X, Y \rangle^2}$
• Ricci curvature $\text{Ric}(X, X)$ trace of the Riemann curvature tensor in the direction of the vector $X$
  • Measures the average sectional curvature of planes containing $X$
• Scalar curvature $S$ contraction of the Ricci curvature with the metric tensor
  • Provides a single real number measuring the overall curvature at each point
• Constant curvature manifolds have the same sectional curvature $K$ at every point and for every 2-plane
  • Examples: Euclidean space ($K=0$), sphere ($K>0$), hyperbolic space ($K<0$)
• Comparison theorems relate the geometry of a Riemannian manifold to that of constant curvature spaces
  • Bonnet-Myers theorem, Synge's theorem, and the Cartan-Hadamard theorem
• Curvature affects the behavior of geodesics, Jacobi fields, and the topology of the manifold
  • Positive curvature implies compact manifolds and converging geodesics
  • Negative curvature implies non-compact manifolds and diverging geodesics

Applications and Real-World Examples

• General relativity models spacetime as a 4-dimensional Lorentzian manifold with a metric tensor
  • Curvature of spacetime related to the presence of matter and energy via Einstein's field equations
• Riemannian geometry used in computer vision and image processing
  • Treats images as 2D manifolds and uses geodesic distances for segmentation and registration
• Machine learning and data analysis utilize Riemannian geometry
  • Manifold learning algorithms (Isomap, LLE) seek to uncover the underlying low-dimensional structure of high-dimensional data
• Robotics and motion planning benefit from Riemannian geometry
  • Configuration spaces of robots modeled as Riemannian manifolds
  • Geodesics used to plan optimal trajectories and avoid obstacles
• Geometric mechanics formulates classical mechanics on Riemannian manifolds
  • Configuration space of a mechanical system treated as a Riemannian manifold with a kinetic energy metric
• Lie groups equipped with bi-invariant Riemannian metrics
  • Geodesics correspond to one-parameter subgroups and the exponential map is the Lie exponential
• Optimization on Riemannian manifolds generalizes classical optimization techniques
  • Gradient descent, conjugate gradient, and trust-region methods adapted to curved spaces
• Shape analysis and computational anatomy study the geometry of shapes and anatomical structures
  • Shapes represented as points on infinite-dimensional Riemannian manifolds of diffeomorphisms or embeddings

Advanced Topics and Further Reading

• Comparison geometry studies Riemannian manifolds with curvature bounds
  • Alexandrov spaces, CAT($k$) spaces, and Gromov-Hausdorff convergence
• Geometric flows evolve Riemannian metrics according to PDEs
  • Ricci flow, mean curvature flow, and Yamabe flow
  • Used to prove the Poincaré conjecture and the geometrization conjecture
• Harmonic maps between Riemannian manifolds generalize the concept of harmonic functions
  • Minimize the energy functional and satisfy the harmonic map equation
• Eigenvalue problems on Riemannian manifolds study the spectrum of the Laplace-Beltrami operator
  • Relate the geometry of the manifold to the properties of eigenfunctions and eigenvalues
• Spin geometry studies Riemannian manifolds with additional structure called a spin structure
  • Allows the construction of spinor bundles and Dirac operators
• Kähler geometry studies complex manifolds with a compatible Riemannian metric
  • Unifies Riemannian, complex, and symplectic geometry
• Finsler geometry generalizes Riemannian geometry by allowing the metric to depend on direction
  • Finsler metrics assign a norm to each tangent space instead of an inner product
• Sub-Riemannian geometry studies manifolds with a distribution of subspaces of the tangent spaces
  • Geodesics are constrained to be tangent to the distribution
• Further reading:
  • "Riemannian Geometry" by Manfredo P. do Carmo
  • "Riemannian Manifolds: An Introduction to Curvature" by John M. Lee
  • "Comparison Geometry" by Karsten Grove
  • "Geometric Flows and Geometric Operators" by Bennett Chow and Dan Knopf