Irreducible holonomy algebras of Riemannian supermanifolds

Possible irreducible holonomy algebras \mathfrakg ì \mathfrakosp(p, q|2m){\mathfrak{g}\subset\mathfrak{osp}(p, q|2m)} of Riemannian supermanifolds under the assumption that \mathfrakg{\mathfrak{g}} is a direct sum of simple Lie superalgebras of classical type and possibly of a 1-dimensional center are classified. This generalizes the classical result of Marcel Berger about the classification of irreducible holonomy algebras of pseudo-Riemannian manifolds.     

曲率与Betti数

本文中，我们首先根据经典的Ricci曲率与Betti数的S．Bochner定理得到了ε－极小Riemann浸入子流形的数量曲率与Betti数的结果。然后，我们考虑了紧致连通Riemann流形中曲率与Betti数之间的关系，推广 了经典的S．Bochner定理。     

完备Riemann流形具有闭的割空间

本文证明了完备的Ｒｉｅｍａｎｎ流形即拥有闭的割空间（ｃｕｔｓｐａｃｅ），这一结论不但完满解答了段海豹在［１］中提出的问题１．６，大大地改进了他的主要结果（［１］，定理１．３），而且作为一个推论，我们还得到了经典Ｂｏｒｓｕｋ－Ｕｌａｍ定理的一个进一步推广．     

Riemannian geometry of Lie algebroids

We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the integrability of Riemannian Lie algebroids.     

两点齐性的 Finsler 流形

该文证明任何一个两点齐性的 Finsler 流形一定是黎曼流形. 证明过程中作者将泛函分析中经典的Mazur 定理推广到不一定是绝对齐次的 Minkowski 空间上.     

两点齐性的Finsler流形

该文证明任何一个两点齐性的Finsler流形一定是黎曼流形．证明过程中作者将泛函分析中经典的Mazur定理推广到不一定是绝对齐次的Minkowski空间上．     

Harnack and mean value inequalities on graphs

We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.     

New examples of Riemannian g.o. manifolds in dimension 7

A Riemannian g.o. manifold is a homogeneous Riemannian manifold (M,g) on which every geodesic is an orbit of a one-parameter group of isometries. It is known that every simply connected Riemannian g.o. manifold of dimension ?5 is naturally reductive. In dimension 6 there are simply connected Riemannian g.o. manifolds which are in no way naturally reductive, and their full classification is known (including compact examples). In dimension 7, just one new example has been known up to now (namely, a Riemannian nilmanifold constructed by C. Gordon). In the present paper we describe compact irreducible 7-dimensional Riemannian g.o. manifolds (together with their “noncompact duals”) which are in no way naturally reductive.     

Geodesic flow on the diffeomorphism group of the circle

We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: the Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds.     

Analytic continuation and identities involving heat,Poisson, wave and Bessel kernels

In this article we use classical formulas involving the K–Bessel function in two variables to express the Poisson kernel on a Riemannian manifold in terms of the heat kernel. We then use the small time asymptotics of the heat kernel on certain Riemannian manifolds to obtain a meromorphic continuation of the associated Poisson kernel to all values of complex time with identifiable singularities. This result reproves in a different setting by different means a well–known theorem due to Duistermaat and Guillemin [DG 75]. Also, we develop analytic expressions for the heat kernel beyond asymptotic expansions. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fiber Bundles and Regular Approximation of Codimension-One Cycles

We derive an approximation of codimension-one integral cycles(and cycles modulo p) in a compact Riemannian manifold bymeans of piecewise regular cycles: we obtain both flat convergence andconvergence of the masses. The theorem is proved by using suitableprincipal bundles with a discrete group. As a byproduct, we give analternative proof of the main results, which does not use the regularitytheory for homology minimizers in a Riemannian manifold. This also givesa result of -convergence.     

Analytic inequalities,isoperimetric inequalities and logarithmic Sobolev inequalities

There is a simple equivalence between isoperimetric inequalities in Riemannian manifolds and certain analytic inequalities on the same manifold, more extensive than the familiar equivalence of the classical isoperimetric inequality in Euclidean space and the associated Sobolev inequality. By an isoperimetric inequality in this connection we mean any inequality involving the Riemannian volume and Riemannian surface measure of a subset α and its boundary, respectively. We exploit the equivalence to give log-Sobolev inequalities for Riemannian manifolds. Some applications to Schrödinger equations are also given.     

The Kernel Theorem of Hilbert-Schmidt operators

All Hilbert-Schmidt operators acting on L²-sections of a vector bundle with fiber a separable Hilbert space H over a compact Riemannian manifold M, are characterized. This is achieved by defining the vector bundle of Hilbert-Schmidt operators on H, and then making use of a classical result known as the Kernel Theorem of Hilbert-Schmidt operators.     

Some smooth Finsler deformations of hyperbolic surfaces

Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectrum rigidity and boundary rigidity all fail to extend to the Finsler category.     

Area preserving transformations in two-dimensional space forms and classical differential geometry

We generalize Scheffers’ method to construct area preserving transformations in the Euclidean plane to Riemannian and Lorentzian two-dimensional space forms in a unified way. We review and extend the classical applications of such transformations in classical differential geometry. We introduce two classes of surfaces in Lorentzian 3-space that admit holomorphic representation and are analogous to the classical Appell and Bonnet surfaces.     

Spectral Flow,Maslov Index and Bifurcation of Semi-Riemannian Geodesics

We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals imply that each nondegenerate and nonnull conjugate (or P-focal)point along a semi-Riemannian geodesic is a bifurcation point.In particular, the semi-Riemannian exponential map is notinjective in any neighborhood of a nondegenerate conjugate point,extending a classical Riemannian result originally due to Morse and Littauer.     

The Bott–Samelson theorem for positive Legendrian isotopies

The classical Bott–Samelson theorem states that if on a Riemannian manifold all geodesics issuing from a certain point return to this point, then the universal cover of the manifold has the cohomology ring of a compact rank one symmetric space. This result on geodesic flows has been generalized to Reeb flows and partially to positive Legendrian isotopies by Frauenfelder–Labrousse–Schlenk. We prove the full theorem for positive Legendrian isotopies.     

A Trudinger–Moser Inequality on a Compact Riemannian Surface Involving Gaussian Curvature

Motivated by a recent work of Chen and Zhu (Commun Math Stat 1:369–385. 2013), we establish a Trudinger–Moser inequality on a compact Riemannian surface without boundary. The proof is based on blow-up analysis together with Carleson–Chang’s result (Bull Sci Math 110:113–127. 1986). This inequality is different from the classical one, which is due to Fontana (Comment Math Helv 68:415–454. 1993), since the Gaussian curvature is involved. As an application, we improve Chen–Zhu’s result as follows: a modified Liouville energy of conformal Riemannian metric has a uniform lower bound, provided that the Euler characteristic is nonzero and the volume of the conformal surface has a uniform positive lower bound.     

An extension of Günther’s volume comparison theorem

The aim of this note if to give an extension of a classical volume comparison theorem for Riemannian manifolds with sectional curvature bounded above (see Günther, P. Einige Sätze über das Volumenelement eines Riemannschen Raumes, Publ. Math. Debrecen 7, 78–93 (1960)). For the case of a n-dimensional simply connected complete Riemannian manifold with nonpositive sectional curvature our theorem states that the function tarea(S _t (p))/t ^n–2 is convex for every pM where S _t (p) denotes the sphere of radius t with center p. In view of area(S ₀ (p))=0, it is easy to see that our theorem implies the classical result. A similar result holds true for simply connected manifolds with sectional curvature bounded above by a negative constant.Research partially supported by Fondecyt Grant # 1000713 and by UTFSM Grant # 120023     

A generalization of the classical sphere theorem

In this paper, we prove a sphere theorem for Riemannian manifolds with partially positive curvature which generalizes the classical sphere theorem.