Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userUniqueness of curvature measures in pseudo-Riemannian geometry
68
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The recently introduced Lipschitz-Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Kunneth-type formula for Lipschitz-Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.
rate research
Read More
The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space $mathrm{Curv}^{mathrm{U}(n)*}$ of dual unitarily invariant curvature measures. Building on the recent results from integral geometry in complex space forms, we describe this algebra structure explicitly as a polynomial algebra. This is a short way to encode all local kinematic formulas. We then characterize the invariant valuations on complex space forms leaving the space of invariant angular curvature measures fixed.
This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, Frechet medians are considered. We prove their statistical consistency and give some quantitative estimations of their robustness with the aid of upper curvature bounds. We also show that, in compact Riemannian manifolds, the Frechet medians of generic data points are always unique. Stochastic and deterministic algorithms are proposed for computing Riemannian p-means. The rate of convergence and error estimates of these algorithms are also obtained. Finally, we apply the medians and the Riemannian geometry of Toeplitz covariance matrices to radar target detection.
In this paper shall we endeavour to substantiate that the evolution of the Riemann- Christoffel tensor or curvature tensor can be expressed entirely by an arbitrary timelike vector field and that the curvature tensor returns to its initial value with respect to change in a particular index. This implies that Poincares recurrence theorem is valid in this cosmological scenario. Also, it has been shown that geodesics can diverge just as they can converge. As is ostensible, this result indicates the existence the of a point of exclusivity - the opposite of a singularity.
We study nice nilpotent Lie algebras admitting a diagonal nilsoliton metric. We classify nice Riemannian nilsolitons up to dimension $9$. For general signature, we show that determining whether a nilpotent nice Lie algebra admits a nilsoliton metric reduces to a linear problem together with a system of as many polynomial equations as the corank of the root matrix. We classify nice nilsolitons of any signature: in dimension $leq 7$; in dimension $8$ for corank $leq 1$; in dimension $9$ for corank zero.
We give a concise proof that large classes of optimal (constant curvature or Einstein) pseudo-Riemannian metrics are maximally symmetric within their conformal class.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
