We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the the orbit space M/G is isometric to a Weyl chamber C which is a Riemannian manifold with corners and certain angle conditions along intersections of faces. We c...

In ([1]), T. Adati and K. Matsumoto defined para-Sasakian and special para-Sasakian manifolds which are considered as special cases of an almost paracontact manifold introduced by I. Sato and K. Matsumoto ([10]). In the same paper, the authors studied conformally symmetric para-Sasakian manifolds and they proved that an ndimensional (n>3) conformally symmetric para-Sasakian manifold is conforma...

This work delves into the space-time theory of 4-dimensional Kaehler manifold. Since isotropic pressure, energy density, and momentum tensor all vanish in a perfect fluid manifold, we have established that this manifold is an Einstein studied equation with cosmological constant it. Finally, demonstrated that, on conformally flat, perfectly velocity vector field infinitesimally spatially isotrop...

The increasing entropy, large-scale isotropy and approximate flatness of the universe are considered in the context of signature change, which is a classical model of quantum tunnelling in quantum cosmology. The signature change hypothesis implies an initial inflationary epoch, the magnetic half of the Weyl curvature hypothesis, and a close analogue of the conformal singularity hypothesis. Addi...

Suppose that (Mn, g) is an asymptotically flat Riemannian spin manifold of positive scalar curvature. The positive mass theorem [1, 2, 3] states that the total mass of the manifold is always positive, and is zero if and only if the manifold is flat. This result suggests that there should be an inequality which bounds the Riemann tensor in terms of the total mass and implies that curvature must ...

Let (Mn, g), n ≥ 4, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given 0 < l ≤ L, we prove that there exists ε = ε(l, L, n) satisfying the following: If the scalar curvature s of g satisfies l ≤ s ≤ L and the Einstein tensor satisfies |Ric − s n g| ≤ ε then M is diffeomorphic to a symmetric space of compact type. This is a smooth analogue of the result...

Vector cross product structures on manifolds include symplectic, volume, G2and Spin (7)-structures. We show that their knot spaces have natural symplectic structures, and we relate instantons and branes in these manifolds with holomorphic disks and Lagrangian submanifolds in their knot spaces. For the complex case, the holomorphic volume form on a Calabi-Yau manifold de…nes a complex vector cro...

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebr...

Lie derivatives arise in studying continuous symmetries of various geometric structures on manifolds. They are also used in symmetry analysis of ordinary and partial differential equations (see [1]). In general relativity the bundle of Weyl spinors SM is a special geometric structure built over the space-time manifold M . The main goal of this paper is to clarify the procedure of applying Lie d...