Lower Semi-Continuity Of Cut Time On Riemannian Manifolds

Introduction

In Riemannian geometry, understanding the behavior of geodesics is crucial for exploring the structure and properties of manifolds. Geodesics, which are locally distance-minimizing curves, play a fundamental role in defining concepts such as distance, curvature, and completeness. Among the key tools for studying geodesics is the cut locus, which marks the farthest points along a geodesic where it remains minimizing. A related concept is the cut time, which quantifies how long one can travel along a geodesic before reaching the cut locus. This article delves into the question of whether the cut time function is lower semi-continuous on the unit tangent bundle of a Riemannian manifold, and explores the implications for the exponential map at the cut point.

Cut time, denoted as c(v), is defined for each unit tangent vector v at a point on the manifold M. It represents the time it takes for the geodesic starting in the direction v to reach the cut locus. In simpler terms, c(v) indicates the farthest distance one can travel along the geodesic in the direction v while still ensuring that the path is the shortest route between its endpoints. The cut locus itself is the set of points reached at the cut time, forming a boundary beyond which the geodesic ceases to be minimizing. Understanding the properties of the cut time function, such as its continuity, is essential for gaining insights into the global structure of the manifold.

The lower semi-continuity of the cut time function has significant implications for the behavior of geodesics and the geometry of the manifold. A function is lower semi-continuous if its value at a point is less than or equal to the limit inferior of its values at nearby points. In the context of cut time, lower semi-continuity ensures that small perturbations in the initial direction v will not cause a sudden jump in the cut time. This property is crucial for various applications, including the study of conjugate points, the injectivity radius, and the topology of the manifold. Furthermore, the behavior of the exponential map at the cut point, exp(c(v)v), is closely related to the cut time function and provides valuable information about the geometry of the cut locus.

In this article, we will explore the question of whether the cut time function is lower semi-continuous, particularly in the context of possibly non-complete Riemannian manifolds. Non-complete manifolds present additional challenges compared to complete manifolds, as geodesics may not be defined for all time. We will examine the conditions under which lower semi-continuity holds and discuss the implications for the exponential map at the cut point. By understanding the properties of cut time and its relationship to the geometry of the manifold, we can gain deeper insights into the structure and behavior of geodesics in Riemannian spaces.

Defining Cut Time and Lower Semi-Continuity

To rigorously address the question of the lower semi-continuity of cut time, it is essential to establish clear definitions for the key concepts involved. This section provides a detailed explanation of cut time, the unit tangent bundle, and lower semi-continuity, laying the foundation for the subsequent analysis.

Cut Time: In Riemannian geometry, the cut time along a geodesic is a crucial concept for understanding the extent to which the geodesic minimizes distance. Consider a Riemannian manifold M and a point p on M. For a unit tangent vector v in the tangent space T_pM, there exists a unique geodesic γ_v(t) starting at p with initial velocity v. The cut time c(v) is defined as the supremum of all times t > 0 such that the geodesic segment γ_v|[0,t] is the unique minimizing geodesic between p and γ_v(t). In simpler terms, c(v) represents the longest time one can travel along the geodesic γ_v(t) while ensuring that the path remains the shortest connection between its endpoints. The point γ_v(c(v)) is called the cut point of p along the geodesic γ_v.

The cut time can be finite or infinite. If c(v) = ∞, the geodesic γ_v(t) is minimizing for all t > 0, implying that there is no cut point along the geodesic. If c(v) < ∞, then γ_v(c(v)) is the cut point, and beyond this point, the geodesic ceases to be the unique minimizer. It is important to note that the cut time can vary continuously or discontinuously with respect to the initial direction v, depending on the geometry of the manifold.

Unit Tangent Bundle: The unit tangent bundle, denoted by SM, plays a crucial role in the study of geodesics. It is defined as the set of all unit tangent vectors on the manifold M. Formally, SM = {(p, v) | p ∈ M, v ∈ T_pM, ||v|| = 1}, where T_pM is the tangent space at the point p, and ||v|| denotes the norm of the vector v induced by the Riemannian metric. The unit tangent bundle is a manifold itself, with a natural projection map π: SM → M that sends each unit tangent vector (p, v) to its base point p. The cut time c can be viewed as a function c: SM → (0, +∞], mapping each unit tangent vector to its corresponding cut time.

The unit tangent bundle provides a convenient setting for studying the behavior of geodesics. Each point in SM represents an initial condition for a geodesic, specifying both the starting point and the initial direction. By considering the cut time as a function on SM, we can analyze its continuity properties and how it varies across different initial conditions.

Lower Semi-Continuity: In the context of real-valued functions, lower semi-continuity is a property that describes the behavior of a function's values as one approaches a particular point. A function f: X → ℝ ∪ {+∞} (where X is a topological space) is said to be lower semi-continuous at a point x ∈ X if

lim inf _y→x f(y) ≥ f(x).

In other words, for any sequence {x_n} in X converging to x, we have

lim inf _n→∞ f(x_n) ≥ f(x).

This means that the function's value at x is less than or equal to the limit inferior of its values at nearby points. Equivalently, f is lower semi-continuous if for any a ∈ ℝ, the set {x ∈ X | f(x) > a} is open in X. This topological definition provides a convenient way to check lower semi-continuity.

In the context of the cut time function c: SM → (0, +∞], lower semi-continuity at a point (p, v) ∈ SM means that for any sequence {(p_n, v_n)} in SM converging to (p, v), we have

lim inf _n→∞ c(p_n, v_n) ≥ c(p, v).

Intuitively, this implies that if we slightly perturb the initial point and direction of a geodesic, the cut time will not suddenly decrease significantly. Lower semi-continuity is a weaker condition than continuity, as it only requires the function to be