Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe spectral geometry of the Weyl conformal tensor
88
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also study similar questions related to the skew-symmetric curvature operator defined by the Weyl conformal curvature tensor.
rate research
Read More
We study the Bochner-Schrodinger operator $H_{p}=frac 1pDelta^{L^potimes E}+V$ on high tensor powers of a positive line bundle $L$ on a symplectic manifold of bounded geometry. First, we give a rough asymptotic description of its spectrum in terms of the spectra of the model operators. This allows us to prove the existence of gaps in the spectrum under some conditions on the curvature of the line bundle. Then we consider the spectral projection of such an operator corresponding to an interval whose extreme points are not in the spectrum and study asymptotic behavior of its kernel. First, we establish the off-diagonal exponential estimate. Then we state a complete asymptotic expansion in a fixed neighborhood of the diagonal.
We prove a Tauberian theorem for singular values of noncommuting operators which allows us to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality. We explain how, via the Birman--Schwinger principle, these asymptotics imply that a semiclassical Weyl law holds for many interesting noncommutative examples. In Connes notation for quantized calculus, we prove that for a wide class of $p$-summable spectral triples $(mathcal{A},H,D)$ and self-adjoint $V in mathcal{A}$, there holds [lim_{hdownarrow 0} h^pmathrm{Tr}(chi_{(-infty,0)}(h^2D^2+V)) = int V_-^{frac{p}{2}}|ds|^p.] where $int$ is Connes noncommutative integral.
Explicit models for the restricted conformal group of the Einstein static universe of dimension greater than two and for its universal covering group are constructed. Based on these models, as an application we determine all oriented and time-oriented conformal Lorentz manifolds whose restricted conformal group has maximal dimension. They amount to the Einstein static universe itself and two countably infinite series of its compact quotients.
Tensor network states provide successful descriptions of strongly correlated quantum systems with applications ranging from condensed matter physics to cosmology. Any family of tensor network states possesses an underlying entanglement structure given by a graph of maximally entangled states along the edges that identify the indices of the tensors to be contracted. Recently, more general tensor networks have been considered, where the maximally entangled states on edges are replaced by multipartite entangled states on plaquettes. Both the structure of the underlying graph and the dimensionality of the entangled states influence the computational cost of contracting these networks. Using the geometrical properties of entangled states, we provide a method to construct tensor network representations with smaller effective bond dimension. We illustrate our method with the resonating valence bond state on the kagome lattice.
In this note we obtain the characterization for asymptotic directions on various subgroups of the diffeomorphism group. We give a simple proof of non-existence of such directions for area-preserving diffeomorphisms of closed surfaces of non-zero curvature. Finally, we exhibit the common origin of the Monge-Ampere equations in 2D fluid dynamics and mass transport.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
