📐Metric Differential Geometry Unit 10 – Jacobi Fields and Conjugate Points

Key Concepts and Definitions

• Jacobi fields represent infinitesimal variations of geodesics in a Riemannian manifold
• Geodesics are curves that minimize distance between points on a manifold and generalize the notion of straight lines in Euclidean space
• Conjugate points occur when a geodesic ceases to be length-minimizing and are closely related to the existence of Jacobi fields
• The Jacobi equation is a second-order linear differential equation that describes the behavior of Jacobi fields along a geodesic
  • Its solutions provide information about the stability and optimality of geodesics
• Riemannian manifolds are smooth manifolds equipped with a Riemannian metric, which allows for the measurement of distances and angles
• Sectional curvature is a geometric invariant that measures the curvature of a Riemannian manifold in two-dimensional subspaces (planes) of the tangent space
• The Levi-Civita connection is a unique, torsion-free, metric-compatible connection on a Riemannian manifold used to define parallel transport and covariant derivatives

Jacobi Fields: Introduction and Significance

• Jacobi fields arise naturally when studying the behavior of nearby geodesics and the stability of geodesic flows
• They provide a way to analyze the infinitesimal variations of geodesics and understand the geometry of the underlying manifold
• Jacobi fields play a crucial role in the development of comparison theorems in Riemannian geometry (Rauch comparison theorem)
• The study of Jacobi fields is essential for understanding the occurrence and properties of conjugate points along geodesics
• Jacobi fields have applications in various areas of differential geometry, including the calculus of variations, optimal control theory, and general relativity
  • In general relativity, Jacobi fields are used to study the stability of geodesics in spacetime and the focusing of light rays (gravitational lensing)

Geometry of Geodesics and Variations

• Geodesics are characterized by the property that their tangent vectors remain parallel when transported along the curve using the Levi-Civita connection
• The exponential map $\exp_p: T_pM \to M$ maps tangent vectors at a point $p$ to points on the manifold by following geodesics emanating from $p$
  • It provides a local diffeomorphism between a neighborhood of the origin in the tangent space and a neighborhood of $p$ on the manifold
• Variations of geodesics are smooth one-parameter families of curves that include a given geodesic (central geodesic)
• The variation field of a geodesic variation is a vector field along the central geodesic that measures the infinitesimal change in the curves of the variation
• The energy functional $E(\gamma) = \frac{1}{2} \int_a^b \lVert \gamma'(t) \rVert^2 dt$ measures the total energy of a curve $\gamma$ and is closely related to the length functional
  • Geodesics are critical points of the energy functional, which leads to the geodesic equation $\nabla_{\gamma'}\gamma' = 0$

Jacobi Equation and Its Solutions

• The Jacobi equation is a second-order linear differential equation that describes the behavior of Jacobi fields along a geodesic $\gamma$
  • It is given by $\nabla_{\gamma'}^2 J + R(J, \gamma')\gamma' = 0$, where $J$ is a Jacobi field, $\nabla$ is the Levi-Civita connection, and $R$ is the Riemann curvature tensor
• Solutions to the Jacobi equation are called Jacobi fields and represent infinitesimal variations of the central geodesic
• The initial conditions for the Jacobi equation consist of a vector $J(0)$ and its covariant derivative $\nabla_{\gamma'}J(0)$ at a point along the geodesic
• The existence and uniqueness of solutions to the Jacobi equation are guaranteed by the linearity of the equation and the smoothness of the Riemannian metric
• The behavior of Jacobi fields is influenced by the curvature of the manifold, as encoded in the Riemann curvature tensor appearing in the Jacobi equation
  • In positively curved spaces (sphere), Jacobi fields tend to converge, while in negatively curved spaces (hyperbolic space), they tend to diverge

Conjugate Points: Theory and Identification

• Conjugate points along a geodesic are points where the exponential map fails to be a local diffeomorphism
• A point $q = \gamma(t_0)$ is conjugate to $p = \gamma(0)$ along the geodesic $\gamma$ if there exists a non-zero Jacobi field $J$ along $\gamma$ such that $J(0) = J(t_0) = 0$
  • Intuitively, conjugate points occur when nearby geodesics emanating from $p$ intersect at $q$
• The multiplicity of a conjugate point is the dimension of the space of Jacobi fields that vanish at both the initial and conjugate points
• Conjugate points are related to the stability and optimality of geodesics
  • A geodesic ceases to be length-minimizing beyond the first conjugate point encountered along its path
• The Jacobi equation and its solutions play a central role in identifying the location and multiplicity of conjugate points
• The Morse index theorem relates the number of conjugate points along a geodesic to the index of the critical point of the energy functional corresponding to the geodesic

Applications in Riemannian Geometry

• Jacobi fields and conjugate points are essential tools in the study of Riemannian geometry and its applications
• The Rauch comparison theorem uses Jacobi fields to compare the behavior of geodesics in a Riemannian manifold with those in a space of constant curvature
  • It provides bounds on the distance between conjugate points and the growth of Jacobi fields based on curvature comparisons
• Jacobi fields are used in the proof of the Hopf-Rinow theorem, which establishes the equivalence between completeness and geodesic completeness for Riemannian manifolds
• The study of Jacobi fields and conjugate points is crucial for understanding the cut locus of a point in a Riemannian manifold
  • The cut locus is the set of points where geodesics emanating from a given point cease to be minimizing
• Jacobi fields play a role in the development of Morse theory on Riemannian manifolds, which relates the topology of the manifold to the critical points of smooth functions
• In general relativity, Jacobi fields are used to analyze the stability of geodesics in spacetime and the focusing of light rays due to gravitational fields (gravitational lensing)

Computational Methods and Examples

• Numerical methods, such as the Runge-Kutta scheme, can be used to solve the Jacobi equation and compute Jacobi fields along geodesics
• Efficient algorithms have been developed to identify conjugate points and the cut locus of a point in Riemannian manifolds
  • These algorithms often rely on the numerical computation of Jacobi fields and the analysis of their behavior
• Software packages, such as GeomLab and Riemannian Manifold Toolkit (RMTK), provide tools for computing geodesics, Jacobi fields, and related geometric quantities
• Examples of Jacobi fields and conjugate points can be explicitly computed in spaces of constant curvature (Euclidean space, sphere, hyperbolic space)
  • In the Euclidean plane, Jacobi fields along straight lines are linear functions, and there are no conjugate points
  • On the sphere, Jacobi fields along great circles vanish at antipodal points, which are conjugate to the initial point
• In more general Riemannian manifolds, such as ellipsoids or surfaces of revolution, Jacobi fields and conjugate points can be analyzed using a combination of analytical and numerical techniques

Connections to Other Areas of Differential Geometry

• The study of Jacobi fields and conjugate points is closely related to various other topics in differential geometry
• Jacobi fields play a role in the calculus of variations and the study of geodesics as critical points of the energy or length functional
• The Morse index theorem, which relates conjugate points to the index of geodesics as critical points, is a key result in Morse theory on Riemannian manifolds
• Jacobi fields and their growth are related to the Ricci curvature and the Laplacian on Riemannian manifolds through comparison theorems (Laplacian comparison theorem)
• The behavior of Jacobi fields is connected to the study of the Riemann curvature tensor and its properties, such as sectional curvature and Ricci curvature
• Jacobi fields and conjugate points have applications in optimal control theory, where they are used to analyze the optimality and stability of control systems on Riemannian manifolds
• In symplectic geometry, Jacobi fields along geodesics of the Sasaki metric on the tangent bundle of a Riemannian manifold are related to the Hamiltonian flow and the geodesic flow