Alphabeta MathJacobi Fields, Conjugate Points, and the Cut Locus
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1 · Prerequisites
Abstract Algebra
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2 · Logical flowchart
3 · Items
Definition
Geodesic variation
Lemma
Covariant derivatives commute up to curvature in a two parameter variation
Definition
Jacobi field
Theorem
Variation field of a geodesic variation is a Jacobi field
Theorem
Existence and uniqueness of jacobi fields from initial data
Corollary
The space of Jacobi fields along a geodesic has dimension two n
Theorem
Every Jacobi field is induced by a geodesic variation
Proposition
Tangential jacobi fields are affine multiples of the velocity
Proposition
Killing fields restrict to Jacobi fields along geodesics
Lemma
Wronskian of two jacobi fields is constant
Theorem
Differential of the exponential map in terms of Jacobi fields
Definition
Conjugate points along a geodesic and their multiplicity
Theorem
Conjugate points are critical values of the exponential map along the geodesic
Proposition
Conjugate instants are isolated unless the geodesic is constant
Proposition
Conjugate points and multiplicity are invariant under affine reparametrization
Theorem
Second variation formula for energy
Definition
Index form of a geodesic segment
Lemma
Integration by parts for the index form
Proposition
Jacobi fields are the null solutions of the index form with fixed endpoints
Theorem
Index lemma
Definition
H-one Riemannian curves and their half-energy
Lemma
Local length comparison for a conjugate-free geodesic
Lemma
H-one local length comparison for a conjugate-free geodesic
Corollary
A geodesic segment before its first conjugate point is locally energy minimizing
Theorem
A geodesic does not minimize past its first conjugate point
Proposition
At a conjugate endpoint the index form is degenerate
Definition
Cut time in a unit tangent direction
Lemma
Minimizing along a geodesic is an initial interval property
Definition
Cut point and cut locus of a point
Lemma
Unit spheres in finite-dimensional normed spaces are sequentially compact
Lemma
The pointwise norm on a tangent space is smooth off the zero vector
Theorem
Characterization of a cut point
Theorem
Cut time is positive and continuous
Corollary
Cut time does not exceed first conjugate time
Proposition
Injectivity radius is the infimum of cut times
Theorem
The cut locus of a point is closed
Theorem
The exponential map is a diffeomorphism on the open tangent cut domain
Theorem
Distance from p is smooth off p and the cut locus
Proposition
Gradient of the distance is the outward unit radial field off the base point and the cut locus
Proposition
Hessian of distance in terms of radial jacobi fields
Theorem
Cut locus of a point has riemannian volume zero
Corollary
Polar integration may discard the cut locus
Remark
The Morse index theorem for geodesics
False statement
Every vector field along a geodesic is a Jacobi field
False statement
Conjugacy is a property of two points independent of the geodesic between them
False statement
A geodesic stops minimizing exactly at its first conjugate point
False statement
The distance from p is smooth on m minus p
False statement
The cut locus of a point is always a smooth hypersurface
False statement
Nullity of the cut locus follows merely because it has empty interior
Example
Jacobi fields in constant sectional curvature
Sources
- John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997), Chapter 10
- Ved Datar, Lectures on Riemannian Geometry (2025), Lectures 21–24
- John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997), Lemma 10.1
- Ved Datar, Lectures on Riemannian Geometry (2025), Definition 21.2.3
- John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997)
- Ved Datar, Lectures on Riemannian Geometry (2025)
- J.-H. Eschenburg, Comparison Theorems in Riemannian Geometry
- Jeffrey M. Lee, Differential and Physical Geometry (draft)
- John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997), Proposition 10.11 and proof
- Ved Datar, Lectures on Riemannian Geometry (2025), Proposition 22.3.1 and proof
- Danny Calegari, Notes on Riemannian Geometry (2015), §5.8, Lemma 5.22
- John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997), Theorem 10.12
- Ved Datar, Lectures on Riemannian Geometry (2025), Theorem 21.1.2
- Ved Datar, Lectures on Riemannian Geometry (2025), Definition 21.2.1
- John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997), Equation (10.15)
- John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997), Proposition 10.14
- Juha Kinnunen, Sobolev Spaces lecture notes
- Zuoqin Wang, Riemannian Geometry (USTC, 2024 Spring), Lecture 20: The index form
- John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997), Theorem 10.15, Corollary 10.13, and Proposition 10.14
- Ved Datar, Lectures on Riemannian Geometry (2025), Proposition 23.1.1(1) and proof
- Gerald Teschl, Topics in Real and Functional Analysis
- Gerald B. Folland, Real Analysis, 2nd ed.