CURVATURE TENSORS IN A LORENTZIAN PARA SASAKIAN MANIFOLD

Abstract

The author and Mishra [1] have introduced some curvature tensors to study their relativistic and geometric properties. Matsumoto and Mihai [2] have introduced the notion of Lorentzian para Sasakian (LP-Sasakian) and studied certain transformations. In this paper some properties of curvature tensors, in a LP-Sasakian manifold, are studied.     

Pappus type theorems for motions along a submanifold

We study the volumes volume(D) of a domain D and volume(C) of a hypersurface C obtained by a motion along a submanifold P of a space form Mⁿ_λ. We show: (a) volume(D) depends only on the second fundamental form of P, whereas volume(C) depends on all the ith fundamental forms of P, (b) when the domain that we move D₀ has its q-centre of mass on P, volume(D) does not depend on the mean curvature of P, (c) when D₀ is q-symmetric, volume(D) depends only on the intrinsic curvature tensor of P; and (d) if the image of P by the ln of the motion (in a sense which is well-defined) is not contained in a hyperplane of the Lie algebra of SO(n−q−d), and C is closed, then volume(C) does not depend on the ith fundamental forms of P for i>2 if and only if the hypersurface that we move is a revolution hypersurface (of the geodesic (n−q)-plane orthogonal to P) around a d-dimensional geodesic plane.     

Geometric connections and geometric Dirac operators on contact manifolds

We construct some natural metric connections on metric contact manifolds compatible with the contact structure and characterized by the Dirac operators they determine. In the case of CR manifolds these are invariants of a fixed pseudo-hermitian structure, and one of them coincides with the Tanaka-Webster connection.     

Intrinsic characterizations for complex hypercylinders and complex hyperspheres

In terms of conditions on the curvature tensors of Riemann-Christoffel, Ricci, Weyl and Bochner we obtain several new characterizations of complex hyperspheres in complex projective spaces, of complex hypercylinders in complex Euclidean spaces and of complex hyperlanes in complex space forms.Aspirant N.F.W.O. (België).     

Maximal surfaces in a certain 3-dimensional homogeneous spacetime

A generalized integral representation formula for spacelike maximal surfaces in a certain 3-dimensional homogeneous spacetime is obtained. This spacetime has a solvable Lie group structure with left invariant metric. The normal Gauß map of maximal surfaces in the homogeneous spacetime is discussed and the harmonicity of the normal Gauß map is studied.     

Conformal deformations of integral pinched 3-manifolds

In this paper we prove that, under an explicit integral pinching assumption between the L²-norm of the Ricci curvature and the L²-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits a conformal metric of positive Ricci curvature. In particular, using a result of Hamilton, this implies that the manifold is diffeomorphic to a quotient of S³. The proof of the main result of the paper is based on ideas developed in an article by M. Gursky and J. Viaclovsky.     

Naturally reductive Riemannian homogeneous structures on some classes of generic submanifolds in complex space forms

We prove that the universal covering spaces of the generic submanifolds of C P _n and of C H _n are naturally reductive homogeneous spaces by determining explicitly tensor fields defining naturally reductive homogeneous structures on them.     

Commutator length of symplectomorphisms

Each element $x$ of the commutator subgroup $[G, G]$ of a group $G$ can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of $x$. The commutator length of $G$ is defined as the supremum of commutator lengths of elements of $[G, G]$. We show that for certain closed symplectic manifolds $(M,\omega)$, including complex projective spaces and Grassmannians, the universal cover $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ of the group of Hamiltonian symplectomorphisms of $(M,\omega)$ has infinite commutator length. In particular, we present explicit examples of elements in $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ that have arbitrarily large commutator length -- the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of $(M,\omega)$. By a different method we also show that in the case $c_1 (M) = 0$ the group $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ and the universal cover ${\widetilde{\Symp}}_0\, (M,\omega)$ of the identity component of the group of symplectomorphisms of $(M,\omega)$ have infinite commutator length.     

Isoperimetric inequalities for special classes of curves

In this paper the classical Banchoff-Pohl inequality, an isoperimetric inequality for nonsimple closed curves in the Euclidean plane, involving the square of the winding number, is sharpened for homothetic or Abresch-Langer solutions of curve shortening. For a larger class of curves and for rotationally symmetric curves, further isoperimetric inequalities containing the rotation number and the winding number, are presented.