pseudo-Riemannian Structures Research Articles

On the cotangent bundle with vertical modified riemannian extensions

Let $M$ be an n-dimensional differentiable manifold with a torsion-free linear connection $\nabla $ which induces on its cotangent bundle ${T^*}M$. The main purpose of the present paper is to study some properties of the vertical modified Riemannian extension on ${T^*}M$ which is given as a new metric in [17]. At first, we investigate a metric connection with torsion on ${T^*}M$. And then, we present the holomorphy properties with respect to a compatible almost complex structure. urthermore, we study locally decomposable Golden pseudo-Riemannian structures on the cotangent bundle endowed with vertical modified Riemannian extension.

Structures of spinors fiber bundles with special relativity of Dirac operator using the Clifford algebra

Abstract The purpose of this article is to demonstrate how to use the mathematics of spinor bundles and their category. We have used the methods of principle fiber bundles obey thorough solid harmonic treatment of pseudo-Riemannian manifolds and spinor structures with Clifford algebras, which couple with Dirac operator to study important applications in cohomology theory.

Pseudo-Riemannian structures in Pati-Salam models

We discuss the role of the pseudo-Riemannian structure of the finite spectral triple for the family of Pati-Salam models. We argue that its existence is a very restrictive condition that separates leptons from quarks, and restricts the whole family of Pati-Salam models into the class of generalized Left-Right Symmetric Models.

On curvature tensors of Norden and metallic pseudo-Riemannian manifolds

AbstractWe study some properties of curvature tensors of Norden and, more generally, metallic pseudo-Riemannian manifolds. We introduce the notion of J-sectional and J-bisectional curvature of a metallic pseudo-Riemannian manifold (M, J, g) and study their properties.We prove that under certain assumptions, if the manifold is locally metallic, then the Riemann curvature tensor vanishes. Using a Norden structure (J, g) on M, we consider a family of metallic pseudo-Riemannian structures {Ja,b}a,b∈ℝ and show that for a ≠ 0, the J-sectional and J-bisectional curvatures of M coincide with the Ja,b-sectional and Ja,b-bisectional curvatures, respectively. We also give examples of Norden and metallic structures on ℝ2n.

INVARIANT PSEUDO-SASAKIAN AND K-CONTACT STRUCTURES ON SEVEN-DIMENSIONAL NILPOTENT LIE GROUPS

This paper studies the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group A complete list of 22 classes of seven-dimensional nilpotent Lie groups which admit pseudo-Riemannian Sasaki structures is found. A list of 25 classes of seven-dimensional nilpotent Lie groups admitting K-contact structures, but not pseudo-Riemannian Sasaki structures, is also presented. All the contact structures considered are central extensions of six-dimensional nilpotent symplectic Lie groups. Formulas that connect the geometric characteristics of six-dimensional nilpotent almost pseudo-Kähler Lie groups and seven-dimensional nilpotent contact Lie groups are established. As is known, for six-dimensional nilpotent pseudo-Kähler Lie groups the Ricci tensor is always zero. In contrast to the pseudo-Kӓhler case, it is shown that on contact seven-dimensional Lie algebras the Ricci tensor is nonzero even in directions of the contact distribution

Homogeneous pseudo-Riemannian structures of linear type

This article is partly a survey and partly a research paper on homogeneous pseudo-Riemannian structures of linear type. In the Riemannian case, these structures furnish characterisations of real, complex and quaternionic hyperbolic spaces. In the Lorentzian case, a related class gives characterisations of singular homogeneous plane waves.

Nonlinear gravitons, null geodesics, and holomorphic disks

We develop a global twistor correspondence for pseudo-Riemannian conformal structures of signature (++--) with self-dual Weyl curvature. Near the conformal class of the standard indefinite product metric on S2×S2, there is an infinite-dimensional moduli space of such conformal structures, and each of these has the surprising global property that its null geodesics are all periodic. Each such conformal structure arises from a family of holomorphic disks in CP3 with boundary on some totally real embedding of RP3 into CP3. Some of these conformal classes are represented by scalar-flat indefinite Kähler metrics, and our methods give particularly sharp results in connection with this special case

Chern-Simons Forms Associated to Homogeneous Pseudo-Riemannian Structures

Forms of Chern-Simons type associated to homogeneous pseudo-Riemannian structures are considered. The corresponding secondary classes are a measure of the lack of a homogeneous pseudo-Riemannian space to be locally symmetric. Explicit computations are done for some pseudo- Riemannian Lie groups and their compact quotients. © 2005 Rocky Mountain Mathematics Consortium.

Homogeneous Lorentzian structures on the oscillator groups

We obtain all the homogeneous pseudo-Riemannian structures on the oscillator groups equipped with a family of left-invariant Lorentzian metrics. Moreover, in the 4-dimensional case we determine all the corresponding reductive decompositions and groups of isometries.

Sections quadratiques fredholmiennes

A Fredholm quadratic section on a Hilbert manifold M is a smooth section of the Banach bundle of symmetric bilinear forms whose associated operators are Fredholm; such a section defines a singular pseudo-Riemannian structure on M. These structures appear naturally after induction on submanifolds of finit codimension in a pseudo-Riemannian Hilbert manifold ( W, Q). The germs of singular pseudo-Riemannian Hilbert structures are classified

Isotropy of semisimple group actions on manifolds with geometric structure

We prove results concerning isotropy of noncompact semisimple Lie group actions that preserve pseudo-Riemannian or affine structures on compact or finite volume manifolds. For example, if a Lie group G acts locally faithfully on a connected compact complete affine manifold M and preserves the affine structure, then the connected component of the identity of each isotropy subgroup is a compact extension of a solvable group. This is a consequence of the result that nontrivial affine actions of connected groups locally isomorphic to SL 2 [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] on such manifolds cannot have fixed points. We also prove that if G is connected semisimple without compact factors and acts isometrically and topologically transitively on a connected finite volume pseudo-Riemannian M , then the action must be (everywhere) locally free. This answers affirmatively a special case of a conjecture by G. D'Ambra and M. Gromov. With only the assumption that no factor of G acts trivially, local freeness is also assured when M is compact and complete.

Reductive homogeneous pseudo-Riemannian manifolds

A classification of homogeneous pseudo-Riemannian structures and a characterization of each primitive class are obtained. Several examples are also given.

Connections on central bimodules in noncommutative differential geometry

We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the different noncommutative versions of differential forms based on derivations. Then we investigate reality conditions and a noncommutative generalization of pseudo-riemannian structures.

Nonflat pseudo-Riemannian space forms and homogeneous pseudo-Riemannian structures of class $S_1$

Nonflat pseudo-Riemannian space forms and homogeneous pseudo-Riemannian structures of class $S_1$

Affinely rigid body and Hamiltonian systems on GL( n, R)

The paper deals with classical Hamiltonian systems with degrees of freedom ruled by the affine group. We discuss Riemannian and pseudo-Riemannian structures on GL( n, R ) invariant under certain physically interpretable subgroups of the transformation group: L(n, R)∋X↦AXB+W, A, B∈GL(n, R), W∈L(n, R) . Hamiltonina systems of the form H= T+ V are analysed, with kinetic terms induced by the mentioned Riemannian structures on GL( n, R ) and with velocity-independent potentials. For physical reasons, special stress is laid on doubly-isotropic models, when H and V are invariant under left and right orthogonal translations, L(n, R)∋X↦AXB, A, B∈O(n, R) . The reduced description of doubly-isotropic dynamics is based on the two-polar decomposition of GL +( n, R ). It turns out that in this representation our doubly-isotropic geodetic problems V=0 can be interpreted in terms of n-element Calogero–Moser and hyperbolic-Sutherland lattices with extra introduced “internal degrees of freedom”, and couplings between lattice points, depending on internal parameters. We discuss certain integrability problems and special solutions generalizing so-called stationary ellipsoids studied in astrophysics. Possibilities of applications in astrophysics, elasticity and in collective or internal dynamics of microobjects are indicated.