Curvature Distribution Vector in Geometry

• Curvature Distribution Vector is a geometric concept that quantifies how curvature is locally and distributionally allocated across surfaces, curves, and convex bodies.
• It represents curvature via vector fields on Riemannian surfaces, vector-valued measures on finite-curvature curves, and explicit formulas on convex bodies.
• This construct underpins integrability conditions and facilitates applications in curvature flow modeling, geometric reconstruction, and non-smooth analysis.

A curvature distribution vector is a geometric construct that encodes the manner in which curvature is distributed in a geometric object. The precise definition and context differ across Riemannian surfaces, curves of finite regularity, and the boundaries of convex bodies. At a fundamental level, it quantifies local and, in certain settings, distributional properties of curvature by means of a vector field or a vector-valued measure. The concept is central in global differential geometry, convex geometry, and the theory of curves with low regularity, providing a rigorous framework for representing curvature flows, reconstructing geometric data, and analyzing integrability and compatibility of geometric structures (Vincze et al., 2020, Mucci et al., 14 Jun 2025, Pereira, 2021).

1. Curvature Distribution Vector for Riemannian Surfaces

On a two-dimensional Riemannian manifold $(M,g)$ with Gauss curvature $K$, a smooth vector field $X \in \Gamma(TM)$ is called a curvature distribution vector if it satisfies

$K = \operatorname{div} X$

where the divergence is taken with respect to the metric $g$. In local coordinates $(u^1, u^2)$, this can be explicitly given by

$$\operatorname{div} X = \frac{1}{\sqrt{\det g}}\ \partial_i(\sqrt{\det g} X^i)$$

so that

$$K(u) = \frac{1}{\sqrt{\det g(u)}} \frac{\partial}{\partial u^i} \left[\sqrt{\det g(u)} X^i(u)\right].$$

The existence of such a vector field on $(M,g)$ is equivalent to the existence of a flat, metric linear connection $\nabla$ with semi-symmetric torsion, specifically of the form $T(X,Y) = \rho(X)Y - \rho(Y)X$, where $\rho$ is a $1$-form dual to $X$ (Vincze et al., 2020).

2. Curvature Distribution as Vector-Valued Measures for Curves

Given a rectifiable curve $\gamma: [0,L] \rightarrow \mathbb{R}^3$ parameterized by arc-length with unit tangent $T(s)$ of bounded variation, the distributional derivative $D T \in \mathcal{M}([0,L]; \mathbb{R}^3)$ is a finite vector-valued Borel measure, with polar decomposition

$D T(s) = \kappa(s)\, d|D T|(s),$

where $|\kappa(s)|=1$ almost everywhere with respect to $|D T|$. The vector measure $\kappa(s) d|D T|(s)$ is the curvature-distribution vector measure, encoding both the magnitude and direction of distributional curvature along the curve, including singularities such as corners (Mucci et al., 14 Jun 2025).

In the smooth setting, this reduces to the classical curvature $k(s)$ and unit normal vector $\kappa(s)=T'(s)/|T'(s)|$, but the vector-valued measure formulation extends the theory to curves with finite total curvature and discontinuities.

3. Curvature Distribution Vector for Convex Bodies

For a convex body $F \subset \mathbb{R}^n$ with boundary point $p \in \partial F$ and a $C^2$ defining function $f$, the curvature in a given tangent direction $u \in T_p F$ is provided by the formula

$$\kappa_F(p, u) = \frac{2\ \langle D^2 f(p) u, u \rangle}{\langle p, \nabla f(p) \rangle \|u\|^2}.$$

By selecting any orthonormal basis $\{u_1, \dots, u_{n-1}\}$ of $T_p F$, the curvature distribution vector is then defined as

$$C(p) = [\kappa_F(p, u_1), \ldots, \kappa_F(p, u_{n-1})] \in \mathbb{R}^{n-1}.$$

This vector characterizes the directional curvature at $p$ and encodes the local osculating quadric structure (Pereira, 2021).

4. Equivalence, Integrability, and Topological Obstructions

On Riemannian surfaces, the divergence representation theorem states that the existence of a curvature distribution vector $X$ ($K = \operatorname{div}\, X$) is equivalent to the existence of a flat, metric connection with prescribed semi-symmetric torsion. This equivalence is central in the local and global integrability of non-Riemannian structures such as generalized Berwald (Finsler) surfaces.

Global obstructions arise: on a compact surface without boundary, the divergence theorem implies $\int_M K\, dA = 0$, which combined with the Gauss–Bonnet theorem gives $\chi(M) = 0$ for the Euler characteristic. Thus, spheres ($S^2$) and other surfaces with nonzero Euler characteristic do not admit a global curvature distribution vector as defined above for the divergence representation (Vincze et al., 2020).

5. Local Expressions and Computation in Frames

In practice, curvature distribution vectors are constructed locally using orthonormal frames or bases of tangent spaces:

• On surfaces, writing $\rho = \rho_1 \omega^1 + \rho_2 \omega^2$ and $X = -\rho^\sharp = -(\rho_1 e_1 + \rho_2 e_2)$, the equation $K = \operatorname{div} X$ is equivalent to a system on the components $(\rho_1, \rho_2)$ involving the connection 1-forms.
• For convex bodies, the curvature vector $C(p)$ is obtained by evaluating the explicit curvature formula in an orthonormal basis, with invariance under orthogonal transformation and change of frame.

Worked examples include explicit computations for the Euclidean plane, hyperbolic plane, ellipses, and ellipsoids, demonstrating how the vector $C(p)$ recovers principal and other normal curvatures (Pereira, 2021, Vincze et al., 2020).

6. Applications in Finsler and Non-Smooth Curve Theory

The existence and construction of a curvature distribution vector underpins the integrability condition for the existence of flat metric connections with semi-symmetric torsion, essential for constructing generalized Berwald structures and non-Riemannian Finsler manifolds. Given a flat, metric connection (with prescribed $X$), parallel transport of Minkowski indicatrices yields Finsler metrics invariant under that connection (Vincze et al., 2020).

In the non-smooth theory of curves, the curvature distribution vector measure $D T$ and its scalar and directional decompositions enable the extension of existence and uniqueness theorems for curves beyond classical $C^2$ regularity. This approach codifies corners and singularities as atoms in the curvature measure and supports reconstruction of curves via measure-theoretic analogs of the moving frame and Frenet–Serret equations (Mucci et al., 14 Jun 2025).

7. Key Properties and Invariance

Curvature distribution vectors exhibit essential covariance properties:

• For convex bodies, rigid motion (actions by $O(n)$) maps the curvature distribution vector consistently: $C(R(p))$ under $F \mapsto R(F)$ is the corresponding rotation of $C(p)$ (Pereira, 2021).
• A change of orthonormal frame in the tangent (or normal) bundle permutes or mixes components of $C(p)$ orthogonally; as one samples a dense set of directions, $C(p)$ captures the complete local curvature structure.
• In principal curvature analysis, extremal values of $C(p)$ correspond to principal curvatures, and in geometric modeling or analysis, $C(p)$ serves as a local feature vector for shape characterization or reconstruction.

These invariance and structural properties facilitate applications in geometric modeling, global analysis, and the extension of differential geometry techniques to broader, less regular contexts (Pereira, 2021, Mucci et al., 14 Jun 2025, Vincze et al., 2020).