Riemannian Manifolds in Which Certain Curvature Operator Has Constant Eigenvalues along Each Circle

Riemannian manifolds for which a natural curvature operator has constant eigenvalues on circles are studied. A local classification in dimensions two and three is given. In the 3-dimensional case one gets all locally symmetric spaces and all Riemannian manifolds with the constant principal Ricci curvatures r ₁ = r ₂ = 0, r ₃= 0 , which are not locally homogeneous, in general.     

Carnot-Caratheodory空间中的变换论

Klein发表著名的埃尔兰根纲领,由群论角度研究了空间变换群的不变量,从而引进了各种不同的几何学．本文利用Felix Klein的观念,研究Carnot-Caratheodory空间{M,Q,g}(又称为次黎曼流形)上的类似问题,给出了次黎曼流形中的共形不变量和射影不变量．本文给出的共形和射影不变量可视为黎曼情形的一种自然推广．由于次黎曼流形与黎曼流形之间有着本质的差异,故此,本文通过次黎曼流形上存在的唯一非完整联络(Nonholonomic connections)来刻画所提的问题．     

Isoperimetric estimates in products

In a product M ₁ × M ₂ of Riemannian manifolds, the least perimeter required to enclose given volume among general regions is at least 1/√ 2 times that among regions of product form, assuming that the isoperimetric profiles of M ₁ and M ₂ are concave. This result sharpens earlier work of Grigor'yan, generalizes results of Bollobás and Leader and of Barthe, and yields a lower bound on the Cheeger isoperimetric constant of a product.     

Sets of permutations that generate the symmetric group pairwise

The paper contains proofs of the following results. For all sufficiently large odd integers n, there exists a set of ²n−1 permutations that pairwise generate the symmetric group Sn. There is no set of ²n−1+1 permutations having this property. For all sufficiently large integers n with n≡2mod4, there exists a set of ²n−2 even permutations that pairwise generate the alternating group An. There is no set of ²n−2+1 permutations having this property.     

Some conformally flat spin manifolds, Dirac operators and automorphic forms

In this paper we study Clifford and harmonic analysis on some examples of conformal flat manifolds that have a spinor structure, or more generally, at least a pin structure. The examples treated here are manifolds that can be parametrized by U/Γ where U is a subdomain of either Sn or Rn and Γ is a Kleinian group acting discontinuously on U. The examples studied here include RPn and the Hopf manifolds S¹×Sn−1. Also some hyperbolic manifolds will be treated. Special kinds of Clifford-analytic automorphic forms associated to the different choices of Γ are used to construct explicit Cauchy kernels, Cauchy integral formulas, Green's kernels and formulas together with Hardy spaces and Plemelj projection operators for Lp spaces of hypersurfaces lying in these manifolds.     

Estimates of covering numbers of convex sets with slowly decaying orthogonal subsets

Covering numbers of precompact symmetric convex subsets of Hilbert spaces are investigated. Lower bounds are derived for sets containing orthogonal subsets with norms of their elements converging to zero sufficiently slowly. When these sets are convex hulls of sets with power-type covering numbers, the bounds are tight. The arguments exploit properties of generalized Hadamard matrices. The results are illustrated by examples from machine learning, neurocomputing, and nonlinear approximation.     

Sobolev-Type Classes of Functions with Values in a Metric Space. II

We prove equivalence of the definitions by the author and by Korevaar and Schoen of the Sobolev classes of mappings of a domain of an arithmetic n-dimensional space to a metric space.     

Second order parabolic equations in Banach spaces with dynamic boundary conditions

In this paper, we exhibit a unified treatment of the mixed initial boundary value problem for second order (in time) parabolic linear differential equations in Banach spaces, whose boundary conditions are of a dynamical nature. Results regarding existence, uniqueness, continuous dependence (on initial data) and regularity of classical and strict solutions are established. Moreover, several examples are given as samples for possible applications.

     

Regularity of Harmonic Functions in Cheeger-Type Sobolev Spaces

We give a geometric approach to the study of the regularity of harmonic functions in Cheeger-type Sobolev spaces, and prove the Hölder continuity of such functions. In the proof, we give a definition of an upper curvature bound of the unit sphere of a Banach space, which seems to be of independent interest.