The goal of this paper is to describe Zermelo’s navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method in order to evaluate its control curvature. We will show that up to changing the Riemannian metric on the manifold the control curvature of Zermelo’s problem has a simple to handle expression which naturally leads to a generalization of the cl...

We give an expression, in terms of the so-called Hermitian intrinsic volumes, for the measure of the set of complex r-planes intersecting a regular domain in any complex space form. Moreover, we obtain two different expressions for the Gauss-BonnetChern formula in complex space forms. One of them expresses the Gauss curvature integral in terms of the Euler characteristic and some Hermitian intr...

To definite and compute differential invariants, like curvatures, for triangular meshes (or polyhedral surfaces) is a key problem in CAGD and the computer vision. The Gaussian curvature and the mean curvature are determined by the differential of the Gauss map of the underlying surface. The Gauss map assigns to each point in the surface the unit normal vector of the tangent plane to the surface...

We construct and investigate smooth orientable surfaces in su(N) algebras. The structural equations of surfaces associated with Grassmannian sigma models on Minkowski space are studied using moving frames adapted to the surfaces. The first and second fundamental forms of these surfaces as well as the relations between them as expressed in the Gauss–Weingarten and Gauss–Codazzi–Ricci equations a...

The goal of this paper is to describe Zermelo’s navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method in order to evaluate its control curvature. We will show that up to change the Riemannian metric on the manifold the control curvature of Zermelo’s problem has a simple to handle expression which naturally leads to a generalization of the clas...

Let A ⊂ R, d ≥ 2, be a compact convex set and let μ = ̺0 dx be a probability measure on A equivalent to the restriction of Lebesgue measure. Let ν = ̺1 dx be a probability measure on Br := {x : |x| ≤ r} equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν = μ◦T−1 and T = φ ·n, where φ : A → [0, r] is a continuous potential with convex sub-level set...