📐Metric Differential Geometry Unit 4 – Geodesics and Distance in Metric Geometry

Key Concepts and Definitions

• Metric space consists of a set $X$ and a distance function $d$ that satisfies certain properties (non-negativity, symmetry, triangle inequality)
• Geodesic defined as a locally length-minimizing curve between two points on a surface or manifold
  • Generalizes the concept of a straight line to curved spaces
• Riemannian manifold is a smooth manifold equipped with a Riemannian metric, enabling the measurement of distances and angles
• Christoffel symbols $\Gamma^k_{ij}$ express the connection coefficients of a Riemannian manifold, used in calculating geodesics
• Exponential map $\exp_p: T_pM \to M$ maps tangent vectors at a point $p$ to points on the manifold along geodesics
• Levi-Civita connection is the unique torsion-free metric connection on a Riemannian manifold, used to define parallel transport and curvature
• Sectional curvature measures the curvature of a Riemannian manifold along two-dimensional subspaces of the tangent space

Metric Spaces and Their Properties

• Metric space $(X, d)$ consists of a set $X$ and a distance function $d: X \times X \to \mathbb{R}$ satisfying:
  • Non-negativity: $d(x, y) \geq 0$ and $d(x, y) = 0$ if and only if $x = y$
  • Symmetry: $d(x, y) = d(y, x)$ for all $x, y \in X$
  • Triangle inequality: $d(x, z) \leq d(x, y) + d(y, z)$ for all $x, y, z \in X$
• Examples of metric spaces include Euclidean space $\mathbb{R}^n$, discrete spaces, and function spaces ($C[a, b]$ with the sup-norm)
• Convergence in metric spaces defined using the distance function: a sequence $(x_n)$ converges to $x$ if $\lim_{n \to \infty} d(x_n, x) = 0$
• Open and closed sets in metric spaces defined using open balls $B(x, r) = \{y \in X : d(x, y) < r\}$
• Completeness is an important property of metric spaces, ensuring the convergence of Cauchy sequences
  • A metric space is complete if every Cauchy sequence converges to a point in the space
• Compact metric spaces are those in which every sequence has a convergent subsequence, equivalent to being complete and totally bounded

Introduction to Geodesics

• Geodesics are curves that locally minimize the distance between points on a manifold or metric space
  • In Euclidean space, geodesics are straight lines, the shortest paths between points
• On a Riemannian manifold $(M, g)$, geodesics are curves $\gamma: I \to M$ 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
• Geodesics are parameterized by arc length, meaning $|\dot{\gamma}(t)| = 1$ for all $t$ in the domain of $\gamma$
• Existence and uniqueness of geodesics guaranteed by the Hopf-Rinow theorem for complete Riemannian manifolds
  • For any point $p \in M$ and tangent vector $v \in T_pM$, there exists a unique geodesic $\gamma$ with $\gamma(0) = p$ and $\dot{\gamma}(0) = v$
• Geodesics are important in understanding the geometry and topology of manifolds, as well as in applications such as general relativity and optimization

Calculating Geodesics in Different Spaces

• In Euclidean space $\mathbb{R}^n$, geodesics are straight lines given by $\gamma(t) = x + tv$, where $x \in \mathbb{R}^n$ and $v \in T_x\mathbb{R}^n \cong \mathbb{R}^n$
• On the unit sphere $S^n$, geodesics are great circles, which can be parameterized using trigonometric functions
  • Example: on $S^2$, a geodesic starting at $(1, 0, 0)$ with initial velocity $(0, \cos\theta, \sin\theta)$ is given by $\gamma(t) = (\cos t, \sin t \cos\theta, \sin t \sin\theta)$
• In hyperbolic space $\mathbb{H}^n$, geodesics are hyperbolic lines or segments, which can be described using hyperbolic trigonometric functions
• On a general Riemannian manifold, geodesics are calculated by solving the geodesic equation $\nabla_{\dot{\gamma}} \dot{\gamma} = 0$
  • This involves using the Christoffel symbols $\Gamma^k_{ij}$ to express the covariant derivative $\nabla_{\dot{\gamma}} \dot{\gamma}$
  • The geodesic equation becomes a system of second-order ODEs in local coordinates, which can be solved numerically or analytically in some cases
• Variational principles, such as the principle of least action, can also be used to derive and calculate geodesics
  • Geodesics minimize the energy functional $E[\gamma] = \frac{1}{2} \int_a^b |\dot{\gamma}(t)|^2 dt$ among all curves with fixed endpoints

Distance Functions and Metrics

• Distance function or metric $d$ on a set $X$ assigns a non-negative real number to each pair of points, satisfying certain properties
  • Distance between a point and itself is always zero: $d(x, x) = 0$ for all $x \in X$
  • Distance is symmetric: $d(x, y) = d(y, x)$ for all $x, y \in X$
  • Triangle inequality: $d(x, z) \leq d(x, y) + d(y, z)$ for all $x, y, z \in X$
• Examples of distance functions include:
  • Euclidean distance in $\mathbb{R}^n$: $d(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}$
  • Manhattan or taxicab distance in $\mathbb{R}^n$: $d(x, y) = \sum_{i=1}^n |x_i - y_i|$
  • Discrete metric on any set $X$: $d(x, y) = 1$ if $x \neq y$ and $d(x, x) = 0$
• On a Riemannian manifold $(M, g)$, the Riemannian distance $d_g(p, q)$ between points $p, q \in M$ is defined as the infimum of the lengths of all piecewise smooth curves connecting $p$ and $q$
  • Length of a curve $\gamma: [a, b] \to M$ is calculated using the Riemannian metric $g$ as $L(\gamma) = \int_a^b \sqrt{g(\dot{\gamma}(t), \dot{\gamma}(t))} dt$
• Riemannian distance satisfies the properties of a metric, making $(M, d_g)$ a metric space
• Geodesics are locally distance-minimizing curves with respect to the Riemannian distance

Curvature and Its Influence on Geodesics

• Curvature measures the deviation of a manifold from being flat or Euclidean
  • Positive curvature corresponds to spaces that curve inward (spheres), while negative curvature corresponds to spaces that curve outward (hyperbolic spaces)
• Riemannian curvature tensor $R(X, Y)Z$ quantifies the curvature of a Riemannian manifold by measuring the non-commutativity of the covariant derivative
  • In local coordinates, the curvature tensor is expressed using the Christoffel symbols and their derivatives
• Sectional curvature $K(X, Y)$ is a scalar quantity that measures the curvature of a manifold along two-dimensional subspaces (sections) of the tangent space
  • Defined as $K(X, Y) = \frac{g(R(X, Y)Y, X)}{|X \wedge Y|^2}$, where $X, Y$ are linearly independent tangent vectors and $|X \wedge Y|^2 = g(X, X)g(Y, Y) - g(X, Y)^2$
• Curvature affects the behavior of geodesics and the global geometry of the manifold
  • In positively curved spaces, geodesics tend to converge, while in negatively curved spaces, they tend to diverge
  • Spaces with constant sectional curvature (e.g., spheres, hyperbolic spaces) have special properties, such as the existence of a unique geodesic between any two points
• Jacobi fields along a geodesic describe the behavior of nearby geodesics and are related to the curvature through the Jacobi equation
  • Jacobi equation: $\nabla_{\dot{\gamma}}^2 J + R(J, \dot{\gamma})\dot{\gamma} = 0$, where $J$ is a Jacobi field along the geodesic $\gamma$

Applications in Physics and Engineering

• General relativity models spacetime as a 4-dimensional Lorentzian manifold, with geodesics representing the paths of free-falling particles and light rays
  • Curvature of spacetime is related to the presence of matter and energy through Einstein's field equations
  • Geodesic deviation and the Jacobi equation are used to study the relative motion of nearby particles in curved spacetime
• Calculus of variations and geodesics are used in optimization problems, such as finding the shortest path between points on a surface
  • Principle of least action in mechanics can be formulated using geodesics on a configuration manifold
  • Geodesic curves on Lie groups are used in control theory and robotics to describe optimal trajectories
• Riemannian geometry and geodesics are applied in machine learning and data analysis
  • Manifold learning techniques, such as Isomap and Locally Linear Embedding, use geodesic distances to uncover the underlying low-dimensional structure of high-dimensional data
  • Geodesic regression extends linear regression to manifold-valued data, using geodesics to model relationships between variables
• Geodesics and curvature are important in the design and analysis of curved surfaces and structures
  • In computer graphics and geometric modeling, geodesics are used for mesh parameterization, texture mapping, and shape analysis
  • Understanding the curvature of surfaces is crucial in the design of thin shell structures, such as aircraft fuselages and architectural facades

Problem-Solving Techniques and Examples

• Calculating geodesics:
  • Given a Riemannian manifold with metric $g$, find the geodesic equation in local coordinates using the Christoffel symbols
  • Solve the resulting system of ODEs with appropriate initial conditions to obtain the geodesic curve
  • Example: on the sphere $S^2$ with metric $g = d\theta^2 + \sin^2\theta d\phi^2$, find the geodesic starting at $(\frac{\pi}{2}, 0)$ with initial velocity $(0, 1)$
• Determining the curvature:
  • Calculate the Christoffel symbols and the Riemannian curvature tensor using the metric in local coordinates
  • Compute the sectional curvature for specific tangent plane sections
  • Example: for the hyperbolic plane $\mathbb{H}^2$ with metric $g = dx^2 + dy^2$, show that the sectional curvature is constant and equal to $-1$
• Applying variational principles:
  • Use the Euler-Lagrange equation to derive the geodesic equation from the energy functional
  • Minimize the energy functional or the length functional to find geodesics between fixed points
  • Example: on a surface with metric $g = E(u, v) du^2 + 2F(u, v) dudv + G(u, v) dv^2$, find the geodesic connecting two points by minimizing the length functional
• Analyzing geodesic properties:
  • Use the exponential map to study the existence and uniqueness of geodesics
  • Investigate the completeness of a manifold using the Hopf-Rinow theorem
  • Example: prove that a compact Riemannian manifold is complete and that any two points can be connected by a minimizing geodesic
• Solving problems in applications:
  • Apply geodesic equations and curvature to model physical phenomena, such as the motion of particles in curved spacetime or the deformation of elastic surfaces
  • Use geodesic distances and curvature to analyze and visualize data on manifolds
  • Example: given a set of points on a sphere, find the shortest path connecting them using geodesics and optimize the path using gradient descent on the sphere