We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a stringmanifold is a per-Hilbert–Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.

Let Fm = (M,F ) be a Finsler manifold and G be the Sasaki– Finsler metric on the slit tangent bundle TM0 = TM {0} of M . We express the scalar curvature ρ̃ of the Riemannian manifold (TM0, G) in terms of some geometrical objects of the Finsler manifold Fm. Then, we find necessary and sufficient conditions for ρ̃ to be a positively homogenenous function of degree zero with respect to the fiber coo...

The aim of this paper is to show that holonomy properties of Finsler manifolds can be very different from those of Riemannian manifolds. We prove that the holonomy group of a positive definite non-Riemannian Finsler manifold of non-zero constant curvature with dimension > 2 cannot be a compact Lie group. Hence this holonomy group does not occur as the holonomy group of any Riemannian manifold. ...

This paper investigates the relationship between two fundamental types of objects associated with a connection on a manifold: the existence of parallel semi-Riemannian metrics and the associated holonomy group. Typically in Riemannian geometry, a metric is specified which determines a Levi-Civita connection. Here we consider the connection as more fundamental and allow for the possibility of se...

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...

The Barzilai-Borwein method, an effective gradient descent method with cleaver choice of the step-length, is adapted from nonlinear optimization to Riemannian manifold optimization. More generally, global convergence of a nonmonotone line-search strategy for Riemannian optimization algorithms is proved under some standard assumptions. By a set of numerical tests, the Riemannian Barzilai-Borwein...

The unit sphere S can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving...

Let  be an n-dimensional Riemannian manifold, and  be its tangent bundle with the lift metric. Then every infinitesimal fiber-preserving conformal transformation  induces an infinitesimal homothetic transformation on .  Furthermore,  the correspondence   gives a homomorphism of the Lie algebra of infinitesimal fiber-preserving conformal transformations on  onto the Lie algebra of infinitesimal ...