The Rauch comparison theorem is a powerful tool in Riemannian geometry. It compares geodesics in different manifolds based on their curvature, providing insights into how curvature affects the spreading of nearby geodesics.

This theorem builds on earlier concepts of sectional curvature and Jacobi fields. It allows us to understand global geometric properties of manifolds by relating them to simpler spaces with constant curvature.

Curvature and Comparison

Sectional Curvature and Comparison Manifolds

• Sectional curvature measures curvature of 2-dimensional planes in tangent space
• Computed using Riemann curvature tensor for orthonormal basis vectors $K(X,Y) = \frac{\langle R(X,Y)Y,X \rangle}{\|X\|^2\|Y\|^2 - \langle X,Y \rangle^2}$
• Provides local geometric information about manifold's shape
• Comparison manifolds serve as reference geometries with constant curvature
  • Includes spheres (positive curvature), Euclidean spaces (zero curvature), and hyperbolic spaces (negative curvature)
• Curvature bounds relate manifold's geometry to comparison manifolds
  • Upper bound $K \leq k$ compares to sphere of radius $1/\sqrt{k}$
  • Lower bound $K \geq k$ compares to hyperbolic space with curvature $k$

Index Form and Variational Approach

• Index form represents second variation of energy functional for geodesics
• Defined for vector fields $V$ along geodesic $\gamma$ as $I(V,V) = \int_0^1 (\|V'\|^2 - \langle R(V,\gamma')\gamma',V \rangle) dt$
• Used to study stability of geodesics and characterize conjugate points
• Variational approach analyzes geodesic behavior through nearby curves
  • Compares length of geodesic to lengths of varied curves
  • Leads to Jacobi equation and study of Jacobi fields

Geodesic Variations

Jacobi Fields and Their Properties

• Jacobi fields describe infinitesimal variations of geodesics
• Satisfy Jacobi equation $J'' + R(J,\gamma')\gamma' = 0$
• Represent tangent vectors to variation of geodesics
• Properties of Jacobi fields
  • Linear space of dimension $2n$ for n-dimensional manifold
  • Uniquely determined by initial conditions $J(0)$ and $J'(0)$
  • Vanishing of Jacobi field indicates conjugate point
• Applications in study of geodesic behavior and manifold geometry

Exponential Map and Its Properties

• Exponential map $\exp_p : T_pM \to M$ sends tangent vectors to geodesic endpoints
• Defined as $\exp_p(v) = \gamma_v(1)$ where $\gamma_v$ is geodesic with initial velocity $v$
• Local diffeomorphism near origin of tangent space
• Properties of exponential map
  • Preserves radial distances from basepoint
  • Jacobi fields along radial geodesics related to differential of exponential map
  • Singularities of exponential map correspond to conjugate points
• Used to define normal coordinates and study local geometry

Singularities

Focal Points and Their Geometric Significance

• Focal points occur where geodesics emanating from submanifold intersect
• Characterized by vanishing Jacobi fields along geodesic
• Geometric significance of focal points
  • Indicate breakdown of distance function's smoothness
  • Related to caustics in optics and wave propagation
  • Affect global geometry and topology of manifold
• Distance to first focal point bounded by curvature (follows from Rauch comparison)
• Examples
  • On sphere, focal points of equator occur at north and south poles
  • In Euclidean space, parallel lines have no focal points

Conjugate Points and Geodesic Behavior

• Conjugate points occur where distinct geodesics with same endpoints intersect
• Characterized by non-trivial Jacobi fields vanishing at both endpoints
• Properties of conjugate points
  • Indicate loss of local minimizing property for geodesics
  • Related to Morse index of energy functional
  • Affect global minimizing properties of geodesics
• Relationship to curvature
  • Positive curvature tends to produce conjugate points (sphere)
  • Negative curvature tends to avoid conjugate points (hyperbolic space)
• Applications in study of cut locus and global geometry of manifold