Sectional curvature and comparison geometry

Along a unit-speed geodesic, let J be a normal Jacobi field with J(0)=0 and a unit initial covariant derivative. Where its norm is nonzero, differentiate the norm twice. The Jacobi equation and sectional curvature K≤0 make the curvature contribution nonnegative in the resulting expression; Cauchy–Schwarz controls the remaining derivative terms. Thus the norm is convex there. Its right derivative at zero is one, so it grows at least as fast as the flat model t.

This also rules out a first positive zero: the lower bound persists up to any proposed first zero and contradicts continuity there. Hence no nontrivial Jacobi field with those normal initial data refocuses. Tangential Jacobi fields with two endpoint zeros are trivial as well, so nonpositive sectional curvature gives no conjugate points. More general Rauch comparisons quantify growth relative to nonzero constant-curvature models, with explicit interval and conjugacy restrictions; this lesson uses only the stated nonpositive case.

Cartan–Hadamard states that a complete connected simply connected Riemannian manifold without boundary and with sectional curvature everywhere at most zero has an exponential map that is a global diffeomorphism at every base point. In particular it is diffeomorphic to Euclidean space, and each pair of points has a unique geodesic segment, which is minimizing. Completeness supplies all initial velocities for all time; curvature prevents local exponential singularities; simple connectedness removes multiple sheets from global topology.

Without simple connectedness, a complete nonpositively curved manifold can have a noninjective exponential map. Its simply connected universal Riemannian cover satisfies the theorem, while distinct lifted endpoints can project to competing routes downstairs. The covering description is useful precisely because it separates local geometry from global identification. Do not replace simple connectedness by connectedness or replace a sectional upper bound with a Ricci lower bound; those are materially different hypotheses.

The unit flat cylinder is the quotient of the Euclidean plane by horizontal translations through 2π. Locally its metric is $d\theta^2+dz^2$ and its Gaussian curvature is zero. Opposite points on one horizontal circle have two shortest routes, corresponding to horizontal displacements +π and −π in the cover. Beyond that distance, continuing one direction around the circle is longer than switching to the other lift. The route has reached a cut point without any local Jacobi refocusing.

Positive constant curvature instead focuses normalized normal fields according to sin(√Kt)/√K, with a first zero at π/√K. Negative constant curvature gives hyperbolic sine and faster-than-linear growth. These exact examples explain comparison directions, but a general theorem also needs its domain and initial normalization. Topological route competition, local exponential derivative degeneracy, and metric-ball volume comparison are related topics with different certificates; one should not infer all three from a single suggestive drawing.