Curvature

Curvature measures how a space bends: it controls triangle angles, geodesic spread, and whether parallel paths converge or split apart.

Why Does This Exist?

Flat-space instincts (parallels never meet, triangle angles sum to 180 degrees, averages are unique) fail on curved model spaces. The failure mode depends on curvature sign: positive curvature pulls geodesics together and can multiply averages, negative curvature spreads them and keeps averages unique. Embedding hierarchies, averaging distributions, and understanding Hessian geometry all need this sign.

Parent pages: manifolds, metrics. Application: hyperbolic geometry.

The one idea you need first

Sectional curvature $K$ assigns a Gaussian-like bend to each tangent plane: $K > 0$ behaves locally like a sphere, $K = 0$ like a plane, $K < 0$ like a saddle. Ricci curvature averages sectionals and controls volume growth.

Think of It Like This

Three ways to draw a triangle

Draw a triangle on a globe through the poles: angles sum past 180 degrees. On paper: exactly 180. On a saddle-shaped chip: below 180, pinched by negative bend.

Same three sides, three different totals. The surface underneath votes on the sum. Algorithms that assume 180 inherit flat bias on curved data.

Where the analogy stops: high-dimensional curvature is a tensor with many sectional values, not one number per space.

How It Actually Works

Jacobi fields measure geodesic spread: $\ddot{J} + K J = 0$ along a unit geodesic. Positive $K$ oscillates (geodesics reconverge, conjugate points appear); negative $K$ grows exponentially (geodesics diverge, no conjugate points, unique Fréchet means). Toponogov-style comparison bounds triangle behavior by model spaces of constant curvature.

Worked example

Triangles with two sides $a = b = 1$ and included angle $C = 90^\circ$ :

Flat: hypotenuse $c = \sqrt{2} \approx 1.414$ . Angles sum $180^\circ$ .
Unit sphere: $c = \arccos(\cos 1 \cos 1) \approx \arccos(0.292) \approx 1.274$ radians ( $\approx 73^\circ$ of arc). Shorter than flat, angles sum $> 180^\circ$ .
Hyperbolic plane ( $K=-1$ ): $\cosh c = \cosh^2 1 \approx 2.381$ , so $c = \text{arcosh}(2.381) \approx 1.513$ . Longer than flat, angles sum $< 180^\circ$ .
Same sides, three lengths ( $1.274$ , $1.414$ , $1.513$ ): sign shows up in every measurement.

Watch Out For

Assuming unique averages on positively curved spaces

Spheres admit multiple Fréchet means for spread data (antipodal clusters tie). Averaging code returns one arbitrarily and downstream layers treat it as canonical.

Check dispersion first; cluster or use medians when data wraps positively curved spaces.

Reading Hessian curvature as manifold curvature

Loss Hessians describe function shape in flat parameter space, not the manifold's intrinsic bend. Conflating them misattributes optimization difficulty to geometry.

Keep the two curvatures labeled separately: loss curvature versus space curvature.

The Quick Version

Curvature sign controls triangle sums, geodesic spread, and mean uniqueness.
Positive curvature reconverges paths and can multiply averages.
Negative curvature spreads paths and keeps means unique, fitting trees well.
Jacobi equation quantifies spread from sectional values.
Hessian shape and manifold bend are different curvatures with different fixes.