Geodesics are curves that locally minimize distance on curved spaces, generalizing straight lines. They're crucial for understanding Riemannian manifolds' geometry, serving as shortest paths between points and satisfying a specific differential equation.

The exponential map connects a manifold's tangent space to the manifold itself. It generates geodesics, preserves vector lengths, and provides a local diffeomorphism. This tool is essential for studying local geometry and geodesic behavior on Riemannian manifolds.

Definition of geodesics

• Geodesics are fundamental objects in Riemannian geometry that generalize the notion of straight lines to curved spaces
• They play a central role in understanding the intrinsic geometry of Riemannian manifolds and serve as the shortest paths between points on the manifold

Geodesics as locally length-minimizing curves

• Geodesics are curves that locally minimize the distance between two points on a Riemannian manifold
• For any point on a geodesic, there exists a neighborhood such that the geodesic segment within that neighborhood is the shortest path connecting its endpoints
• This property distinguishes geodesics from other curves on the manifold and highlights their importance in defining the intrinsic geometry

Geodesic equation

• The geodesic equation is a second-order differential equation that characterizes geodesics on a Riemannian manifold
• In local coordinates, the geodesic equation is given by:

$\frac{d^2x^i}{dt^2} + \Gamma^i_{jk}\frac{dx^j}{dt}\frac{dx^k}{dt} = 0$

where $x^i$ are the coordinates of the curve, $t$ is an affine parameter along the curve, and $\Gamma^i_{jk}$ are the Christoffel symbols of the Levi-Civita connection

• Solving the geodesic equation with appropriate initial conditions yields the geodesics on the manifold

Uniqueness of geodesics

• Given a point on a Riemannian manifold and a tangent vector at that point, there exists a unique geodesic passing through the point with the given tangent vector
• This uniqueness property ensures that geodesics are well-defined and can be used to study the geometry of the manifold
• The uniqueness of geodesics is a consequence of the existence and uniqueness theorem for ordinary differential equations applied to the geodesic equation

Geodesics on Riemannian manifolds

• On a Riemannian manifold, geodesics are determined by the metric tensor, which encodes the geometric properties of the manifold
• The metric tensor induces a Levi-Civita connection, which is used to define the geodesic equation
• Examples of geodesics on specific Riemannian manifolds include:
  • Great circles on a sphere
  • Straight lines on Euclidean space
  • Hyperbolic geodesics on hyperbolic space

Exponential map

• The exponential map is a fundamental tool in Riemannian geometry that relates the tangent space at a point to the manifold itself
• It provides a way to generalize the notion of exponential function to curved spaces and plays a crucial role in studying geodesics and the local geometry of Riemannian manifolds

Definition of exponential map

• For a point $p$ on a Riemannian manifold $M$ and a tangent vector $v$ in the tangent space $T_pM$, the exponential map $\exp_p: T_pM \to M$ is defined as follows:
  • $\exp_p(v)$ is the point on $M$ obtained by following the unique geodesic starting at $p$ with initial velocity $v$ for a unit time
• The exponential map takes a tangent vector and maps it to a point on the manifold, establishing a local diffeomorphism between the tangent space and the manifold

Exponential map as geodesic flow

• The exponential map can be interpreted as the geodesic flow on the tangent bundle of the manifold
• For each tangent vector $v$ at a point $p$, the exponential map traces out the geodesic starting at $p$ with initial velocity $v$
• This interpretation highlights the role of the exponential map in generating geodesics and studying their properties

Properties of exponential map

• The exponential map has several important properties:
  • It is a local diffeomorphism near the origin of the tangent space
  • It preserves the length of tangent vectors: $d(p, \exp_p(v)) = |v|$, where $d$ is the Riemannian distance and $|v|$ is the norm of the tangent vector
  • It is a radial isometry: the exponential map preserves the length of radial geodesics emanating from the base point
• These properties make the exponential map a powerful tool for studying the local geometry of Riemannian manifolds and understanding the behavior of geodesics

Injectivity radius of exponential map

• The injectivity radius of the exponential map at a point $p$ is the largest radius $r$ such that the exponential map is a diffeomorphism from the open ball of radius $r$ in the tangent space $T_pM$ to an open neighborhood of $p$ on the manifold
• The injectivity radius measures the size of the largest geodesically convex neighborhood around a point
• It provides information about the local injectivity of the exponential map and the existence of conjugate points along geodesics

Geodesics and topology

• Geodesics play a crucial role in understanding the topological properties of Riemannian manifolds
• The behavior of geodesics, such as their completeness and the existence of conjugate points, is closely related to the global topology of the manifold

Geodesically convex neighborhoods

• A subset $U$ of a Riemannian manifold $M$ is called geodesically convex if for any two points $p,q \in U$, there exists a unique geodesic segment connecting $p$ and $q$ that is entirely contained in $U$
• Geodesically convex neighborhoods are important for local analysis on Riemannian manifolds, as they ensure the uniqueness of geodesics and the well-definedness of the exponential map
• The existence of geodesically convex neighborhoods is related to the injectivity radius of the exponential map

Geodesic completeness

• A Riemannian manifold is said to be geodesically complete if every geodesic can be extended indefinitely in both directions
• Geodesic completeness is a global property that ensures the existence of geodesics between any two points on the manifold
• It has important implications for the topology and geometry of the manifold, such as the Hopf-Rinow theorem

Hopf-Rinow theorem

• The Hopf-Rinow theorem is a fundamental result in Riemannian geometry that establishes the equivalence between several important properties of Riemannian manifolds:
  • Geodesic completeness
  • Completeness as a metric space with respect to the Riemannian distance
  • The existence of a point such that the exponential map at that point is surjective
• The theorem highlights the deep connection between the geodesic structure, the metric structure, and the topology of Riemannian manifolds

Hadamard manifolds

• A Hadamard manifold is a simply connected, complete Riemannian manifold with non-positive sectional curvature
• Hadamard manifolds have several important properties related to geodesics:
  • They are diffeomorphic to Euclidean space
  • The exponential map at any point is a global diffeomorphism
  • Any two points can be connected by a unique geodesic
• Examples of Hadamard manifolds include the Euclidean space, the hyperbolic space, and certain symmetric spaces

Geodesic distance

• The geodesic distance is a metric on a Riemannian manifold that measures the length of the shortest geodesic connecting two points
• It is a fundamental concept in Riemannian geometry and plays a crucial role in studying the metric and topological properties of Riemannian manifolds

Definition of geodesic distance

• For any two points $p,q$ on a Riemannian manifold $M$, the geodesic distance $d(p,q)$ is defined as the infimum of the lengths of all piecewise smooth curves connecting $p$ and $q$
• If $\gamma: [a,b] \to M$ is a geodesic connecting $p$ and $q$, then the geodesic distance is given by:

$d(p,q) = \int_a^b \sqrt{g_{\gamma(t)}(\gamma'(t),\gamma'(t))} dt$

where $g$ is the Riemannian metric tensor and $\gamma'(t)$ is the tangent vector to the geodesic at $t$

Metric space structure induced by geodesic distance

• The geodesic distance induces a metric space structure on the Riemannian manifold
• It satisfies the axioms of a metric:
  • Non-negativity: $d(p,q) \geq 0$ and $d(p,q) = 0$ if and only if $p=q$
  • Symmetry: $d(p,q) = d(q,p)$
  • Triangle inequality: $d(p,q) \leq d(p,r) + d(r,q)$ for any points $p,q,r$ on the manifold
• The metric space structure allows for the study of topological properties of the manifold, such as completeness and compactness, using tools from metric geometry

Relationship between geodesic and Riemannian distance

• The geodesic distance and the Riemannian distance, defined using the Riemannian metric tensor, are closely related
• In general, the geodesic distance between two points is greater than or equal to the Riemannian distance
• The two distances coincide for nearby points, where the exponential map is a local isometry
• However, for points that are far apart, the geodesic distance may be strictly greater than the Riemannian distance due to the curvature of the manifold

Cut locus and conjugate points

• The cut locus of a point $p$ on a Riemannian manifold is the set of points $q$ for which the geodesic from $p$ to $q$ is no longer minimizing beyond $q$
• Conjugate points are points along a geodesic where the exponential map fails to be a local diffeomorphism
• The presence of cut points and conjugate points affects the behavior of geodesics and the properties of the geodesic distance
• Understanding the cut locus and conjugate points is important for studying the global geometry and topology of Riemannian manifolds

Applications of geodesics

• Geodesics have numerous applications in various areas of mathematics and physics, showcasing their importance in understanding geometric and physical phenomena

Shortest paths on surfaces

• Geodesics provide a natural way to define shortest paths on surfaces
• On a smooth surface equipped with a Riemannian metric, geodesics are the curves that minimize the distance between points
• Finding geodesics on surfaces has practical applications in computer graphics, robotics, and navigation systems
• Examples include finding the shortest path between two points on a sphere (great circles) or on a terrain surface

Geodesics in general relativity

• In general relativity, geodesics play a fundamental role in describing the motion of particles and the propagation of light in curved spacetime
• Freely falling particles and light rays follow geodesics in the spacetime manifold, which is equipped with a pseudo-Riemannian metric
• The geodesic equation in general relativity incorporates the effects of gravity and provides a geometric description of the motion of objects in the presence of gravitational fields
• Studying geodesics in general relativity helps in understanding gravitational lensing, the orbits of planets and satellites, and the behavior of light near massive objects

Geodesic curvature and Gauss-Bonnet theorem

• Geodesic curvature measures the deviation of a curve from being a geodesic on a surface
• It is an important concept in the study of curves on surfaces and has applications in differential geometry and topology
• The Gauss-Bonnet theorem relates the total geodesic curvature of a closed curve on a surface to the integral of the Gaussian curvature over the enclosed region
• This theorem establishes a deep connection between the intrinsic geometry of a surface and its topological properties, such as the Euler characteristic
• Applications of the Gauss-Bonnet theorem include classifying surfaces, studying the global geometry of surfaces, and understanding the behavior of geodesics on surfaces

Geodesic flow and dynamical systems

• The geodesic flow is a dynamical system on the tangent bundle of a Riemannian manifold that describes the motion of particles along geodesics
• It is defined using the exponential map and provides a way to study the long-term behavior of geodesics on the manifold
• The properties of the geodesic flow, such as ergodicity and mixing, provide insights into the geometry and dynamics of the manifold
• Studying the geodesic flow has applications in classical mechanics, statistical mechanics, and the theory of dynamical systems
• Techniques from dynamical systems, such as the study of Lyapunov exponents and entropy, can be used to analyze the behavior of geodesics and understand the complexity of the geodesic flow on Riemannian manifolds