Geodesic convexity and curvature effects

Work on an open strongly geodesically convex domain U: any two points have a unique minimizing geodesic lying entirely in U. A function is geodesically convex when its value along that constant-speed geodesic stays below the linear interpolation of endpoint values. Strong convexity subtracts a positive multiple of the squared endpoint distance times t(1−t). The domain condition avoids an ambiguous choice among multiple shortest routes and avoids a route leaving the region where the objective or assumptions are defined.

For a twice differentiable function on this open domain, nonnegative intrinsic Hessian quadratic forms are equivalent to geodesic convexity by differentiating along geodesics. The first-order inequality uses the logarithm vector from x to y, with its uniqueness ensured by the domain. A critical point is consequently a global minimum within U, and strong convexity gives uniqueness there. These conclusions do not assert optimality outside U. Restricting to an arbitrary coordinate patch without checking the geodesics is insufficient, because straight coordinate segments do not generally represent intrinsic minimizing paths.

On a complete simply connected manifold with nonpositive sectional curvature, the Hadamard theorem makes the exponential at each point globally invertible, providing unique minimizing geodesics. This simplifies the global domain issue, but it does not make every Euclidean rate proof valid without checking curvature-dependent distance comparisons or smoothness constants. On positively curved spaces, cut loci and nonunique geodesics can obstruct the same global constructions. Small strongly convex neighborhoods remain useful, provided iterates are kept inside them.

A smooth geodesically convex function on the whole connected compact manifold without boundary must be constant: a maximum exists, its differential vanishes, and convexity makes that critical maximum a global minimum as well. Thus a nonconstant smooth sphere objective cannot be globally geodesically convex on the entire sphere. Convexity claims for sphere problems must restrict the domain or weaken the conclusion. Curvature assumptions, domain size and injectivity restrictions are therefore part of a theorem’s content rather than optional background about the space.

On the positive real line, equip each tangent with gₓ(ξ,η)=ξη/x². The coordinate u=log x is an isometry to the ordinary real line, so geodesics interpolate log coordinates linearly and all positive reals form one strongly convex domain. Squared distance to a positive reference a is therefore half the square of log(x/a), a globally strongly geodesically convex objective. Its Euclidean second derivative can change sign, illustrating that ambient coordinate convexity and intrinsic convexity are distinct properties.

The metric gradient is x log(x/a), and the exponential map multiplies x by exp(ξ/x). An exponential gradient step consequently multiplies the log error by one minus the step size. In this particular flat logarithmic coordinate, familiar quadratic convergence calculations apply exactly. A normalization retraction or a different metric would define another algorithm. This solvable model offers a useful benchmark for implementations, while its one-dimensional zero curvature should not be used to infer curvature-free global rates for general matrix manifolds.