Exponentially Small Splitting for the Pendulum: A Classical Problem Revisited

In this paper, we study the classical problem of the exponentially small splitting of separatrices of the rapidly forced pendulum. Firstly, we give an asymptotic formula for the distance between the perturbed invariant manifolds in the so-called singular case and we compare it with the prediction of Melnikov theory. Secondly, we give exponentially small upper bounds in some cases in which the perturbation is bigger than in the singular case and we give some heuristic ideas how to obtain an asymptotic formula for these cases. Finally, we study how the splitting of separatrices behaves when the parameters are close to a codimension-2 bifurcation point.     

About non-coincidence of invariant manifolds and intrinsic low dimensional manifolds (ILDM)

The present paper contains an analysis of some aspects of a well known method of Intrinsic Low-Dimensional Manifolds (ILDM), which is regularly used for model reduction purposes in a number of combustion problems. One of these aspects relates to an existence of additional solutions (so-called “ghost”-manifolds), which represent intrinsic low-dimensional manifolds and do NOT represent any slow invariant manifold even for two-dimensional singularly perturbed systems (for a small but finite singular parameter). These “ghost”-manifolds are examples that contradict to the conjecture about the coincidence of ILDM and slow invariant manifolds published previously. Another aspect of the ILDM-method concerns the so-called transition zones (turning manifolds) between different invariant manifolds. It is shown that transition manifolds can not be correctly described by the ILDM-method. This statement is illustrated by an example taken from the mathematical theory of combustion.     

Exotic torus manifolds and equivariant smooth structures on quasitoric manifolds

In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent torus manifolds which are not homeomorphic. Moreover, we characterize those groups which appear as the fundamental groups of locally standard torus manifolds. In the second part we give a classification of quasitoric manifolds and certain six-dimensional torus manifolds up to equivariant diffeomorphism. In the third part we enumerate the number of conjugacy classes of tori in the diffeomorphism group of torus manifolds. For torus manifolds of dimension greater than six there are always infinitely many conjugacy classes. We give examples which show that this does not hold for six-dimensional torus manifolds.     

Atiyah—Patodi—Singer type index theorems for manifolds with splitting of $ \eta $-invariants

We construct self-adjoint extensions of Dirac operators on manifolds with corners of codimension 2, which generalize the Atiyah—Patodi—Singer boundary conditions. The boundary conditions are related to geometric constructions, which convert problems on manifolds with corners into problems on manifolds with boundary and wedge singularities. In the case, where the Dirac bundle is a super-bundle, we prove two general index theorems, which differ by the splitting formula for -invariants. Further we work out the de Rham, signature and twisted spin complex in closer detail. Finally we give a new proof of the splitting formula for the -invariant. Submitted: October 1999, Revised version: March 2001.     

Point leaf maximal singular Riemannian foliations in positive curvature

We generalize the notion of fixed point homogeneous isometric group actions to the context of singular Riemannian foliations. We find that in some cases, positively curved manifolds admitting these so-called point leaf maximal SRF's are diffeo/homeomorphic to compact rank one symmetric spaces. In all cases, manifolds admitting such foliations are cohomology CROSSes or finite quotients of them. Among non-simply connected manifolds, we find examples of such foliations which are non-homogeneous.     

THE DIFFERENTIAL INTEGRAL EQUATIONS ON SMOOTH CLOSED ORIENTABLE MANIFOLDS

1 IntroductionSillce tl1e limit value fOrlnula, viz. tl1e Plemelj fOrn1ula, of the Cauthe type integraJ withBochner-Martinelli kernel was proved in 1957[1], it has beell successfully used to the study Ofsingular i1ltegral equatious, solvi11g the 0b--equation, holomorphic extension, 0--closed exten-sion and C-R 111al1ifolds[2-51. Evideutly, the researcl1 of higher order singular integrals withBochuer-Martinelli kerllel itself also l1as important significallce. In 1952, J. Hadanmrd firstde…     

Morse theory and Lyusternik-Shnirelman theory in geometric control theory

Questions, related to the application of the ideas of global analysis to optimal control problems, are considered. A theory of Lyusternik-Shnirelman type is constructed for Hilbert manifolds with singularities, the so-called transversally convex subsets. Conditions for the nondegeneracy of the critical points (the extremal controls) are established in the optimal control problem, related to a smooth control system of constant rank, and a formula for their Morse index is given.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 39, pp. 41–117, 1991.     

A Cheeger–Müller theorem for Cappell–Miller torsions on manifolds with boundary

In this paper, we extend the Cappell–Miller analytic torsion to manifolds with boundary under the absolute and relative boundary conditions and using the techniques of Brüning-Ma and Su-Zhang, we get the anomaly formula of it for odd dimensional manifolds. Then by the methods of Brüning-Ma, Cappell–Miller and Su-Zhang, we get the Cheeger–Müller theorem for the Cappell–Miller analytic torsion on odd dimensional manifolds with boundary up to a sign. As a consequence of the main theorem, we get the gluing formula for the Cappell–Miller analytic torsion which generalizes a theorem of Huang.     

A Groshev Type Theorem for Convergence on Manifolds

We deal with Diophantine approximation on the so-called non-degenerate manifolds and prove an analogue of the Khintchine–Groshev theorem. The problem we consider was first posed by A. Baker [1] for the rational normal curve. The non-degenerate manifolds form a large class including any connected analytic manifold which is not contained in a hyperplane. We also present a new approach which develops the ideas of Sprindzuk"s classical method of essential and inessential domains first used by him to solve Mahler"s problem [28].     

Melnikov函数和Poincaré映射

本文中我们给出了Melnikov函数和Poincaré映射的关系,从而给出了Melnikov方法的新的证明.本文的优点是给出了更明确的解,并把次谐分支的Melnikov函数与稳定流形与不稳定流形横截相交的Melnikov函数统一成为一个公式.     

Curvature vs. Almost Hermitian Structures

A Bochner-type formula for almost Hermitian manifolds is introduced. From this formula, one can find obstructions imposed by the curvature to the existence of certain almost Hermitian structures on compact manifolds.     

Absolute curvature measures,II

The absolute curvature measures for sets of positive reach in R ^d introduced in [7] satisfy the following kinematic relations: Their integrated values on the intersections with (or on the tangential projections onto) uniformly moved p-planes are constant multiples of the corresponding absolute curvature measures of the primary set. In the special case of convex bodies the first result is the so-called Crofton formula. An analogue for signed curvature measures is well known in the differential geometry of smooth manifolds, but the motion of absolute curvatures used there does not lead to this property. For the special case of smooth compact hypermanifolds our absolute curvature measures agree with those introduced by Santaló [4] with other methods.In the appendix, the section formula is applied to motion invariant random sets.     

General formula for lower bound of the first eigenvalue on Riemannian manifolds

A general formula for the lower bound of the first eigenvalue on compact Riemannian manifolds is presented. The formula improves the main known sharp estimates including Lichnerowicz’ s estimate and Zhong-Yang’s estimate. Moreover, the results are extended to the noncompact manifolds. The study is based on the probabilistic approach (i.e. the coupling method).     

KOPPELMAN-LERAY FORMULA ON COMPLEX MANIFOLDS

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Second variation and Legendrian stabilities of minimal Legendrian submanifolds in Sasakian manifolds

First, we derive a new second variation formula which holds for minimal Legendrian submanifolds in Sasakian manifolds. Using this, we prove that any minimal Legendrian submanifold in an η-Einstein Sasakian manifold with “nonpositive” η-Ricci constant is stable. Next we introduce the notion of the Legendrian stability of minimal Legendrian submanifolds in Sasakian manifolds. Using our second variation formula, we find a general criterion for the Legendrian stability of minimal Legendrian submanifolds in η-Einstein Sasakian manifolds with “positive” η-Ricci constant.     

The Künneth formula in Floer homology for manifolds with restricted contact type boundary

We prove the Künneth formula in Floer (co)homology for manifolds with restricted contact type boundary. We use Viterbo's definition of Floer homology, involving the symplectic completion by adding a positive cone over the boundary. The Künneth formula implies the vanishing of Floer (co)homology for subcritical Stein manifolds. Other applications include the Weinstein conjecture in certain product manifolds, obstructions to exact Lagrangian embeddings, existence of holomorphic curves with Lagrangian boundary condition, as well as symplectic capacities. Supported by ENS Lyon, école Polytechnique (Palaiseau) and ETH (Zürich).     

Log geometry and exploded manifolds

Log Gromov-Witten invariants have recently been defined separately by Gross and Siebert and Abramovich and Chen. This paper provides a dictionary between log geometry and holomorphic exploded manifolds in order to compare Gromov-Witten invariants defined using exploded manifolds or log schemes. The gluing formula for Gromov-Witten invariants of exploded manifolds suggests an approach to proving analogous gluing formulas for log Gromov-Witten invariants.     

On structure-preserving connections

In this paper we find the formula of connections under which an almost complex structure is covariantly constant. These types of connections on anti-Kähler–Codazzi manifolds are described. Also, twin metric-preserving connections are analyzed for quasi-Kähler manifolds. Finally, anti-Hermitian Chern connections are investigated.     

Characterizations of umbilic hypersurfaces in warped product manifolds

We consider the closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using an integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.     

Twisted signatures of flag manifolds

A product formula for some twisted signatures of flag manifolds is proved. The result is used to compute twisted signatures of some flag manifolds from those of Grassmannians, and by that to deduce some upper bounds of the stable span.