Riemannian Geometry Unit 7 Review: Jacobi Fields and the Exponential Map

Key Concepts and Definitions

• Riemannian manifold: A smooth manifold equipped with a Riemannian metric, which is a positive definite inner product on each tangent space
• Geodesic: A curve that locally minimizes the distance between points on a Riemannian manifold, generalizing the concept of a straight line in Euclidean space
  • Geodesics are characterized by the property that their tangent vectors remain parallel when transported along the curve
• Exponential map: A map that takes a tangent vector at a point on a Riemannian manifold and maps it to the point obtained by following the geodesic starting at the original point in the direction of the tangent vector for a unit time
• Jacobi field: A vector field along a geodesic that describes the infinitesimal variation of the geodesic under a family of geodesics
  • Jacobi fields satisfy the Jacobi equation, a second-order linear differential equation
• Sectional curvature: A measure of the curvature of a Riemannian manifold at a point in a given tangent plane, obtained by comparing the behavior of geodesics emanating from that point to those in Euclidean space

Geometric Intuition

• Riemannian geometry extends the concepts of Euclidean geometry to curved spaces, where the notion of distance is determined by the Riemannian metric
• Geodesics generalize the concept of straight lines to curved spaces, representing the shortest paths between points on the manifold
• The exponential map provides a way to map tangent vectors to points on the manifold, allowing for the construction of geodesics and the exploration of the manifold's local geometry
  • Intuitively, the exponential map "wraps" the tangent space onto the manifold, preserving the direction of the tangent vectors
• Jacobi fields describe how nearby geodesics spread out or converge, providing insight into the manifold's curvature
  • In positively curved spaces (spheres), geodesics tend to converge, while in negatively curved spaces (hyperbolic spaces), they tend to diverge
• Sectional curvature quantifies the curvature of the manifold in a given tangent plane, with positive curvature corresponding to spherical geometry and negative curvature corresponding to hyperbolic geometry

Mathematical Foundations

• Riemannian metrics are symmetric, positive-definite $(0,2)$-tensors that define an inner product on each tangent space of the manifold
  • In local coordinates, the Riemannian metric is represented by a symmetric, positive-definite matrix $g_{ij}$
• The Levi-Civita connection is a unique, torsion-free connection on a Riemannian manifold that is compatible with the metric
  • It allows for the definition of parallel transport and the covariant derivative of vector fields
• Geodesics are curves $\gamma(t)$ that satisfy the geodesic equation: $\nabla_{\dot{\gamma}}\dot{\gamma} = 0$, where $\nabla$ is the Levi-Civita connection and $\dot{\gamma}$ is the tangent vector to the curve
• The Riemann curvature tensor is a $(1,3)$-tensor that measures the non-commutativity of the covariant derivative and provides a complete description of the manifold's curvature
  • In local coordinates, the Riemann curvature tensor is given by $R^i_{jkl} = \partial_k \Gamma^i_{jl} - \partial_l \Gamma^i_{jk} + \Gamma^i_{mk}\Gamma^m_{jl} - \Gamma^i_{ml}\Gamma^m_{jk}$, where $\Gamma^i_{jk}$ are the Christoffel symbols of the Levi-Civita connection
• The sectional curvature is related to the Riemann curvature tensor by $K(X,Y) = \frac{R(X,Y,Y,X)}{|X|^2|Y|^2 - \langle X,Y \rangle^2}$, where $X$ and $Y$ are linearly independent tangent vectors

Jacobi Fields: Theory and Applications

• Jacobi fields arise from variations of geodesics and satisfy the Jacobi equation: $\nabla_{\dot{\gamma}}^2 J + R(J, \dot{\gamma})\dot{\gamma} = 0$, where $J$ is the Jacobi field, $\gamma$ is the geodesic, and $R$ is the Riemann curvature tensor
• Jacobi fields can be interpreted as infinitesimal deformations of geodesics, describing how nearby geodesics spread out or converge
• The behavior of Jacobi fields is closely related to the curvature of the manifold
  • In positively curved spaces, Jacobi fields tend to converge, while in negatively curved spaces, they tend to diverge
  • In flat spaces (Euclidean spaces), Jacobi fields maintain a constant separation
• Jacobi fields have applications in the study of geodesic deviation, which measures how nearby geodesics spread out or converge
  • The geodesic deviation equation is given by $\frac{D^2 \xi^i}{dt^2} + R^i_{jkl}\dot{\gamma}^j\dot{\gamma}^k\xi^l = 0$, where $\xi$ is the deviation vector field and $\frac{D}{dt}$ is the covariant derivative along the geodesic
• Jacobi fields also play a role in the study of conjugate points, which are points along a geodesic where nearby geodesics intersect
  • The existence of conjugate points is related to the singularities of the exponential map and has implications for the global geometry of the manifold

The Exponential Map Explained

• The exponential map $\exp_p: T_pM \to M$ at a point $p$ on a Riemannian manifold $M$ maps tangent vectors to points on the manifold
  • For a tangent vector $v \in T_pM$, $\exp_p(v)$ is defined as the point reached by following the geodesic starting at $p$ with initial velocity $v$ for a unit time
• The exponential map is a local diffeomorphism near the origin of the tangent space, meaning that it provides a local parameterization of the manifold by geodesics
• The differential of the exponential map at the origin is the identity map, $d(\exp_p)_0 = \text{id}_{T_pM}$, which reflects the fact that geodesics initially follow the direction of their tangent vectors
• The exponential map is not necessarily injective or surjective globally, and its singularities are related to the presence of conjugate points along geodesics
• The exponential map can be used to define normal coordinates around a point, which provide a local coordinate system in which geodesics through the point correspond to straight lines in the coordinate space
  • Normal coordinates are useful for studying the local geometry of the manifold and for performing calculations in a simplified setting

Connections to Curvature

• The behavior of Jacobi fields and the exponential map is closely tied to the curvature of the Riemannian manifold
• In positively curved spaces (spheres), Jacobi fields along geodesics tend to converge, leading to the existence of conjugate points and the breakdown of the exponential map's injectivity
  • This reflects the fact that geodesics on a sphere converge at the antipodal points
• In negatively curved spaces (hyperbolic spaces), Jacobi fields along geodesics tend to diverge, and the exponential map is typically a global diffeomorphism
  • This reflects the fact that geodesics in hyperbolic spaces diverge exponentially and do not intersect again
• The Riemann curvature tensor provides a complete description of the manifold's curvature and determines the behavior of Jacobi fields through the Jacobi equation
• The sectional curvature, which is a scalar quantity derived from the Riemann curvature tensor, measures the curvature of the manifold in a given tangent plane
  • Positive sectional curvature corresponds to spherical geometry, while negative sectional curvature corresponds to hyperbolic geometry
• The relationship between curvature, Jacobi fields, and the exponential map has important implications for the global geometry of the manifold, such as the existence of conjugate points, the completeness of geodesics, and the presence of cut loci

Practical Examples and Calculations

• Spheres (positively curved):
  • On the unit sphere $S^2$, geodesics are great circles, and the exponential map at a point $p$ maps a tangent vector $v$ to the point reached by following the great circle starting at $p$ in the direction of $v$ for a distance equal to the length of $v$
  • Jacobi fields along geodesics on $S^2$ converge at the antipodal points, leading to the existence of conjugate points and the breakdown of the exponential map's injectivity
• Hyperbolic spaces (negatively curved):
  • In the Poincaré disk model of the hyperbolic plane $\mathbb{H}^2$, geodesics are circular arcs perpendicular to the boundary of the disk, and the exponential map at a point $p$ maps a tangent vector $v$ to the point reached by following the geodesic starting at $p$ in the direction of $v$ for a distance equal to the hyperbolic length of $v$
  • Jacobi fields along geodesics in $\mathbb{H}^2$ diverge exponentially, and the exponential map is a global diffeomorphism
• Calculating sectional curvature:
  • For a surface embedded in $\mathbb{R}^3$ with metric induced by the Euclidean metric, the sectional curvature at a point $p$ is given by the Gaussian curvature $K = \frac{eg-f^2}{EG-F^2}$, where $E,F,G$ and $e,f,g$ are the coefficients of the first and second fundamental forms, respectively
  • For a Riemannian manifold with metric given in local coordinates, the sectional curvature can be calculated using the formula $K(X,Y) = \frac{R(X,Y,Y,X)}{|X|^2|Y|^2 - \langle X,Y \rangle^2}$, where $R$ is the Riemann curvature tensor expressed in terms of the Christoffel symbols

Advanced Topics and Extensions

• Riemannian submersions: Maps between Riemannian manifolds that preserve the length of horizontal tangent vectors, allowing for the study of the relationship between the geometry of the domain and codomain manifolds
  • The O'Neill tensors $A$ and $T$ describe the curvature of the fibers and the integrability of the horizontal distribution, respectively
• Comparison theorems: Results that relate the geometry of a Riemannian manifold to that of a space of constant curvature (sphere, Euclidean space, or hyperbolic space) under certain curvature assumptions
  • The Rauch comparison theorem relates the behavior of Jacobi fields to the curvature of the manifold, while the Toponogov comparison theorem relates the angle and distance properties of triangles
• Morse theory: The study of the relationship between the topology of a manifold and the critical points of smooth functions defined on it
  • The Morse index of a critical point is related to the behavior of the Hessian of the function, which can be interpreted as a Jacobi field along a geodesic
• Symmetric spaces: Riemannian manifolds with a high degree of symmetry, characterized by the property that the geodesic reflection at any point is an isometry
  • Symmetric spaces have a rich algebraic structure and are closely related to Lie groups and their homogeneous spaces
• Infinite-dimensional Riemannian geometry: The study of Riemannian manifolds modeled on infinite-dimensional Hilbert spaces, arising in areas such as the geometry of loop spaces and the study of diffeomorphism groups
  • Many of the concepts and techniques from finite-dimensional Riemannian geometry, such as geodesics, curvature, and Jacobi fields, can be extended to the infinite-dimensional setting, although there are also significant differences and challenges