The exponential map is a powerful tool in Riemannian geometry, linking tangent spaces to the manifold itself. It maps vectors to points, creating a bridge between linear and curved spaces. This connection allows us to study local properties of manifolds using familiar linear algebra techniques.

Normal coordinates, derived from the exponential map, simplify calculations around a point on the manifold. They make the metric tensor look like the identity matrix and eliminate Christoffel symbols at that point. This simplification is crucial for understanding curvature and geodesics locally.

Exponential Map and Normal Coordinates

Definition and Properties of Exponential Map

• Exponential map $\exp_p: T_pM \to M$ maps tangent vectors to points on the manifold
• Assigns to each vector $v \in T_pM$ the point $\gamma_v(1)$, where $\gamma_v$ represents the geodesic with initial velocity $v$
• Provides a way to "flatten" a neighborhood of a point on the manifold onto the tangent space
• Preserves lengths of vectors along radial geodesics
• Maps straight lines through the origin in $T_pM$ to geodesics on $M$

Normal Coordinates and Their Applications

• Normal coordinates form a local coordinate system around a point $p$ on the manifold
• Obtained by composing the inverse of the exponential map with a linear isomorphism
• Simplify calculations and geometric interpretations in a neighborhood of $p$
• Christoffel symbols vanish at $p$ in normal coordinates
• Metric tensor $g_{ij}$ at $p$ becomes the identity matrix in normal coordinates
• Useful for studying local properties of the manifold (curvature, geodesics)

Local Diffeomorphism and Its Implications

• Exponential map is a local diffeomorphism in a neighborhood of the origin in $T_pM$
• Ensures the existence of normal coordinates around any point on the manifold
• Implies the existence of a star-shaped neighborhood around $p$ where $\exp_p$ is injective
• Allows for the transfer of geometric properties between the tangent space and the manifold
• Crucial for defining and studying concepts like the injectivity radius

Geodesics and Gauss Lemma

Gauss Lemma and Its Geometric Interpretation

• Gauss Lemma states that geodesics emanating from a point are orthogonal to distance spheres
• Formalizes the intuition that geodesics are "straight lines" on the manifold
• Implies that the exponential map preserves inner products of radial and transverse vectors
• Crucial for understanding the geometry of normal neighborhoods
• Helps prove that the exponential map is a local isometry

Properties of Radial Geodesics

• Radial geodesics are geodesics emanating from a fixed point $p$
• Correspond to straight lines through the origin in the tangent space $T_pM$
• Minimize distance between $p$ and points along their path
• Form a foliation of a normal neighborhood around $p$
• Play a key role in the construction of normal coordinates

Injectivity Radius and Its Significance

• Injectivity radius at a point $p$ measures the size of the largest ball on which $\exp_p$ is injective
• Determines the extent to which the exponential map provides a faithful representation of the manifold
• Relates to the curvature and global geometry of the manifold
• Influences the behavior of geodesics and the existence of conjugate points
• Important in the study of the cut locus and global properties of Riemannian manifolds

Global Properties

Hopf-Rinow Theorem and Its Consequences

• Hopf-Rinow theorem connects completeness, geodesic completeness, and global properties of manifolds
• States that for a connected Riemannian manifold, the following are equivalent:
  1. The manifold is complete as a metric space
  2. Closed and bounded subsets are compact
  3. The manifold is geodesically complete
• Implies that any two points on a complete, connected Riemannian manifold can be joined by a minimizing geodesic
• Ensures the existence of geodesics realizing the distance between any two points
• Has important applications in the study of global geometry and topology of manifolds

Applications and Extensions of Hopf-Rinow

• Provides a criterion for determining when a Riemannian manifold is geodesically complete
• Allows for the extension of local properties to global ones in complete manifolds
• Crucial in the study of the cut locus and conjugate points on complete manifolds
• Generalizes to other geometric contexts (Finsler geometry, sub-Riemannian geometry)
• Forms the basis for further results in global Riemannian geometry (Cartan-Hadamard theorem)