The exponential map and normal neighborhoods

Try

On the unit sphere, a geodesic starting at p with tangent vector v of length r reaches exp_p(v)=cos r p+sin r v/r at time one, with the zero vector interpreted by continuity. For small v this map provides coordinates around p. At r=π every direction reaches the antipode, so global injectivity fails.

The exponential map packages geodesic initial-value solutions. In general exp_p(v)=γ_v(1), wherever the geodesic with initial velocity v exists to time one. Near zero, local ODE theory guarantees this definition.

Predict

Predict why the derivative of exp at zero is the identity. The curve t↦exp_p(tv) is the geodesic with initial velocity v, so its derivative at zero is v.

Predict whether local invertibility is enough to compare every possible connecting path. A competitor may leave the coordinate neighborhood. A minimizing proof must control that possibility rather than only optimize paths staying inside the chart.

Experiment

Compare radial behavior.

Surface	Radial geodesic from p	First global warning
Plane	p+tv	No wrapping
Sphere	Great-circle motion	Antipodal coincidence
Cylinder	Unrolled straight line, then wrap	Competing routes

For the sphere formula, take the dot product with p to obtain cos r. This checks both the endpoint angle and the fact that the tangent term is orthogonal to p. At r=0, use the limit sin r/r=1 instead of dividing by zero.

Understand

Smooth dependence of geodesics on initial data makes exp smooth. The scaling relation exp_p(tv)=γ_v(t) shows d(exp_p)_0(v)=v. The inverse function theorem therefore yields a normal neighborhood.

Gauss’s lemma establishes radial orthogonality. Let F(r,a)=exp_p(r v(a)) with ||v(a)||=1. Write T=F_r and J=F_a. The radial curves are unit-speed geodesics. Differentiate ⟨T,J⟩ in r: ∂r⟨T,J⟩=⟨D_rT,J⟩+⟨T,D_rJ⟩. The first term is zero. Torsion-free interchange gives D_rJ=D_aT, so the second term is (1/2)∂a||T||²=0. At r=0, J=0. Hence the pairing remains zero.

In normal polar coordinates the radial and angular directions are orthogonal, with unit radial coefficient. A path’s speed is therefore at least |r′|. Its length from the center to radius ρ is at least ρ, and the radial segment attains that bound.

Apply

To handle competitors leaving the coordinate patch, choose a normal ball of radius R whose closure lies inside the normal chart and a target with radius ρ<R. A path that leaves this ball must first reach radial coordinate R, costing at least R>ρ before departure. Thus it cannot improve on the radial segment. Paths staying in the ball obey the radial lower bound. This proves genuine local minimizing behavior, not merely a constrained optimization inside coordinates.

Repair “The exponential map is a global diffeomorphism.” Local ODE and inverse-function results do not exclude later intersections, conjugate points or incomplete geodesics. The sphere’s antipode is an explicit failure.

For optional depth, equality in the radial length bound forces no angular motion away from the center, giving uniqueness of the short minimizing radial segment in the chosen normal ball.

Continue

Distinguish geodesic existence, normal coordinates, local minimality and global injectivity. Continue to the Gauss equation to derive curvature directly from the metric connection.