In this study we introduce a new tensor in semi-Riemannian manifold, named the M*-projective curvature which generalizes m-projective tensor. We start by deducing some fundamental geometric properties of After that, pseudo symmetric manifolds (PM?S)n. A non-trivial example has been used to show existence such manifold. series interesting conclusions. establish, among other things, that if scala...

Abstract: Completing the results achieved in a previous paper, we prove the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients in smooth D-dimensional Lorentzian manifolds. To this end, it is shown that, in any Lorentzian manifold, a sort of “local Wick rotation” of the metric can be performed provided the metric is a locally analytic function of the coordinates and the coordinates ar...

This is a survey of recent contributions to the area of special Kähler geometry. It is based on lectures given at the 21st Winter School on Geometry and Physics held in Srni in January 2001. 1 Remarkable features of special Kähler manifolds A (pseudo-) Kähler manifold (M,J, g) is a differentiable manifold endowed with a complex structure J and a (pseudo-) Riemannian metric g such that (i) J is ...

We search for pseudo-differential operators acting on holomorphic Sobolev spaces. The should mirror the standard mapping property in analogues. setting is a closed real-analytic Riemannian manifold, or Lie group with bi-invariant metric, and spaces are defined fixed Grauert tube about core manifold. find that every operator commutant of Laplacian this kind. Moreover, so all certain analytic ope...

We study a (relativistic) Wiener process on complexified (pseudo-)Riemannian manifold. Using Nelson’s stochastic quantization procedure, we derive three equivalent descriptions for this problem. If the has purely real quadratic variation, obtain one-sided that is encountered in theory of Brownian motion. In case, result coincides with Feyman–Kac formula. On other hand, imaginary two-sided mecha...

Conformally Stäckel manifolds can be characterized as the class of $n$-dimensional pseudo-Riemannian $(M,G)$ on which Hamilton–Jacobi equation $$ G(\nabla u, \nabla u) = 0 for null geodesics and Laplace $-\Delta\_G , \psi 0$ are solvable by R-separation variables. In particular case in metric has Riemannian signature, they provide explicit examples metrics admitting a set $n-1$ commuting confor...

Morse theory relates the set of critical points of a smooth functional defined on a Hilbert manifold to the topology of the manifold itself. Morse himself gave the first application of his theory to Riemannian geometry (cf. [6, 11, 12]), proving two very nice and famous results. In order to recall them, consider a Riemannian manifold (M, 〈 · , · 〉x) with Riemannian structure 〈 · , · 〉x. A curve...

Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. Recently, B.-Y. Chen studied Lagrangian submanifolds in complex space forms which are ideal. He proved that such submanifolds are minimal. He also classified ideal Lagrangian submani...

The complete homogeneous pseudo-Riemannian manifolds of constant non-zero curvature were classified up to isometry in 1961 [1]. In the same year, a structure theory [2] was developed for complete fiat homogeneous pseudo-Riemannian manifolds. Here that structure theory is sharpened to a classification. This completes the classification of complete homogeneous pseudo-Riemannian manifolds of arbit...

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...