A basic question, first addressed by A. Lichnerowicz, is to characterize the pseudo-Riemannian manifolds (M,σ) for which Conf M does not preserve any metric in [σ]; in this case, Conf M is essential. The Lichnerowicz conjecture, proved by J. Lelong-Ferrand [LF1], says, for (M,σ) a Riemannian manifold of dimension ≥ 2, if Conf M is essential, then M is conformally equivalent to the round sphere ...

let (m,g ) be a compact immersed hypersurface of (rn+1,) , λ1 the first nonzeroeigenvalue, α the mean curvature, ρ the support function, a the shape operator, vol (m ) the volume of m,and s the scalar curvature of m. in this paper, we established some eigenvalue inequalities and proved theabove.1) 1 2 2 2 2m ma dv dvn∫ ρ ≥ ∫ α ρ ,2)( )2 2 1 2m 1 mdv s dvn nα ρ ≥ ρ∫ − ∫ ,3) if the scalar curvatu...

There is a general method, applicable in many situations, whereby a pseudo–Riemannian metric, invariant under the action of some Lie group, can be deformed to obtain a new metric whose geodesics can be expressed in terms of the geodesics of the old metric and the action of the Lie group. This method applied to Euclidean space and the unit sphere produces new examples of complete Riemannian metr...

Quantum chaotic states over a noncommutative monoid, a unitalization of a noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are described in terms of their infinitely divisible generating functionals over the simple monoid-valued fields on an atomless ‘space-time’ set. A canonical decomposition of the logarithmic conditionally posive-definite generating functional is c...

From [4] we continue the algebraic investigation of generalized and equiaffine curvature tensors in a given pseudo-Euclidean vector space and study different orthogonal, irreducible decompositions in analogy to the known decomposition of algebraic curvature tensors. We apply the decomposition results to characterize geometric properties of Codazzi structures and relative hypersurfaces; particul...

Biconservative submanifolds, with important role in mathematical physics and differential geometry, arise as the conservative stress-energy tensor associated to variational problem of biharmonic submanifolds. Many examples biconservative hypersurfaces have constant mean curvature. A famous conjecture Bang-Yen Chen on Euclidean spaces says that everybiharmonic submanifold has null Inspired by co...

of Prof. Yau's lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) A geometric structure on a manifold is a cover by coordinate systems [a " sub-atlas " ] in which the transition functions from one coordinate system to another are not arbitrary mappings but rather belong to some specific set of mappings from (open subsets of) Euclidean space to ...

The Euclidean action with acceleration has been analyzed in Ref. 1, and referred to henceforth as Paper I, for its Hamiltonian and path integral. In this paper, the state space of the Hamiltonian is analyzed for the case when it is pseudo-Hermitian (equivalent to a Hermitian Hamiltonian), as well as the case when it is inequivalent. The propagator is computed using both creation and destruction...

Given a geometric structure on $\mathbb{R}^{n}$ with $n$ even (e.g. Euclidean, symplectic, Minkowski, pseudo-Euclidean), we analyze the set of points inside domain definition an arbitrary given $\mathcal{C}^1$ vector field, where value field equals left/right gradient--like some fixed $\mathcal{C}^2$ potential function, although non-integrability condition holds at each such point. Particular e...