We generalize the Zermelo navigation on Riemannian manifolds (M,h), admitting a space dependence of a ship's speed 0 < |u(x)|h ≤ 1 in the presence of a perturbation W̃ determined by a strong (critical) velocity vector eld satisfying |W̃ (x)|h = |u(x)|h, with application of Finsler metric of Kropina type.

Analogue black holes in non-Riemannian effective spacetime of moving vortical plasmas described by moving magnetohydrodynamic (MHD) flows. This example is an extension of acoustic torsion recently introduced in the literature (Garcia de Andrade,PRD(2004),7,64004), where now the presence of artificial black holes in moving plasmas is obtained by the presence of an horizon in the non-Riemannian s...

In 1923, Eisenhart 1 gave the condition for the existence of a second-order parallel symmetric tensor in a Riemannian manifold. In 1925, Levy 2 proved that a second-order parallel symmetric nonsingular tensor in a real-space form is always proportional to the Riemannian metric. As an improvement of the result of Levy, Sharma 3 proved that any second-order parallel tensor not necessarily symmetr...

In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlinear Schrödinger equations on some Riemannian manifolds like the standard 2-sphere S and the hyperbolic 2-space H(−1). Using the similar idea, we establish such blow-up results on higher dimensional standard sphere and hyperbolic n-space. Extensions to n-dimensional Riemanni...

Sixty years ago, S. B. Myers and N. E. Steenrod (Ann. of Math. 40 (1939), 400-416) showed that the isometry group of a Riemannian manifold without boundary has a stru ture of Lie group. Re ently A. V. Bagaev and N. I. Zhukova (Siberian Math. J. 48 (2007), 579-592) proved the same result for a Riemannian orbifold. In this paper, we rstly show that the isometry group of a Riemannian manifold M wi...

We define a manifold M where objects c ∈ M are curves, which we parameterize as c : S → lR (n ≥ 2, S is the circle). We study geometries on the manifold of curves, provided by Sobolev–type Riemannian metrics H . These metrics have been shown to regularize gradient flows used in Computer Vision applications, see [13, 16] and references therein. We provide some basic results of H metrics; and, fo...

Let (M, g) be a compact Riemannian manifold with universal covering M̃ . The simplest asymptotic invariant which can be associated to (M, g) is the exponential growth rate h(g) of volume on the universal covering, called the volume entropy. If B r (p) denotes the geodesic ball of radius r about p ∈ M̃ and volg(B r (p)) describes its volume with respect to the Riemannian metric g lifted to M̃ then ...

Theorem 1. Suppose (M; g) is a noncompact complete Riemannian manifold with Ric (n 1), we have 0 (n 1) =4. The estimate is sharp since the spectrum is the ray [(n 1) =4;+1) for the hyperbolic space H. For Kähler manifolds, this estimate can be improved. On a Kähler manifold (M; g) of complex dimension n, where g is the Riemannian metric, let ! = g (J ; ) be the Kähler form. In local holomorphic...

Let (M, g) be a four dimensional compact Riemannian manifold without boundary, (N, h) ⊂ R be a compact Riemannian submanifold without boundary. We establish the existence of a global weak solution to the heat flow of extrinsic biharmonic maps from M to N , which is smooth away from finitely many singular times. As a consequence, we prove that if Π4(N) = {0}, then any free homotopy class α ∈ [M,...

For a bounded domain Ω equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from (Ω, g) to a compact Riemannian manifold (N, h) ↪→ Rk without boundary. We generalize the notion of stationary harmonic maps and prove their partial regularity. We also discuss the global Lipschitz and piecewise C1,α-regularity of harmonic maps from (Ω, g) to manifolds that su...