Manifold Theorem Research Articles

A Cheeger–Müller theorem for manifolds with wedge singularities

We study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, we give uniform constructions of the resolvent and heat kernel on suitable manifolds with corners. When the wedge manifold and the base of the wedge are odd dimensional, this is used to obtain a Cheeger-M\"uller theorem relating analytic torsion with the Reidemeister torsion of the natural compactification by a manifold with boundary.

A de Rham decomposition type theorem for contact sub-Riemannian manifolds

In this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that (M, H, g) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field xi is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point qin M such that the holonomy group Psi (q) acts reducibly on H(q) yielding a decomposition H(q) = H_1(q)oplus cdots oplus H_m(q) into Psi (q)-irreducible factors. Using parallel transport we obtain the decomposition H = H_1oplus cdots oplus H_m of H into sub-distributions H_i. Unlike the Riemannian case, the distributions H_i are not integrable, however they induce integrable distributions Delta _i on M/xi , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that T(U/xi )=Delta _1oplus cdots oplus Delta _m, and the latter decomposition of T(U/xi ) induces the decomposition of U/xi into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.

Vanishing Theorems for Riemannian Manifolds with Nonnegative Scalar Curvature and Weighted p-Poincaré Inequality

In this paper, we give some vanishing theorems for harmonic p-forms on complete noncompact Riemannian manifolds satisfying a weighted p-Poincare inequality with nonnegative scalar curvature and under pointwise curvature pinching conditions which are bounded from above by the weight function.

Twisted dirac operators and Kastler-Kalau-Walze theorems for six-dimensional manifolds with boundary

In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.

A Robust Direct Parameter Identification of Exponentially Damped Low-Frequency Oscillation in Power Systems

Low-frequency oscillations in power systems can be modeled as an exponentially damped sinusoid (EDS) signal. Its frequency, damping factor, and amplitude are identified by the robust algorithm proposed in this paper. Under the condition of no noise, the exponentially convergent property of the proposed identification is proved by the use of time scale change, variable transformation, slow integral manifold, averaging method, and Lyapunov stability theorem in sequence. Under the condition of bounded additive noise, the antinoise performance of the identification of each parameter is investigated by the perturbed system theorem and error synthesis principle. The robustness of the proposed method is embodied in the following aspects: the exponential convergence for EDS signal with a wide range of frequency, especially with a rather low frequency; the boundary values of identification errors resulting from high-frequency sinusoidal noise of both frequency and damping factor can be adjusted by tuning the design parameters; and the different effects of the four design parameters on tracking performance and antinoise performance of each parameter identification. Simulation results demonstrate the performance of the algorithm and validate the conclusions.

Comparison Theorems in Finsler Geometry with Integral Weighted Ricci Curvature Bounds and Their Applications

We establish the Laplacian comparison theorems and volume comparison theorems for Finsler manifolds under integral weighted Ricci curvature bounds. As their applications, we obtain some results on integral weighted Ricci curvature and topology for Finsler manifolds. We also obtain some results on the upper bound of first eigenvalue for Finsler manifolds under integral weighted Ricci curvature bounds.

Eigenvalue comparison theorems for Finsler manifolds with integral Ricci curvature bound

Eigenvalue comparison theorems for Finsler manifolds with integral Ricci curvature bound

On compatible linear connections with totally anti-symmetric torsion tensor of three-dimensional generalized Berwald manifolds

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory (Vincze in J AMAPN 21:199–204, 2005) such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann–Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection is uniquely determined by its torsion tensor. If the torsion is zero then we have a classical Berwald manifolds. Otherwise the torsion is a strange data we need to express in terms of quantities of the Finsler manifold. In the paper we are going to give explicit formulas for the linear connections with totally anti-symmetric torsion tensor of three-dimensional generalized Berwald manifolds (Theorem 2). The results are based on averaging of (intrinsic) Finslerian quantities by integration over the indicatrix surfaces. They imply some consequences for the base manifold as a Riemannian space with respect to the averaged Riemannian metric (Theorems 3 and 4). The possible cases are Riemannian spaces of constant zero curvature, constant positive curvature or Riemannian spaces admitting Killing vector fields of constant Riemannian length.

A Robust Online Identification of Sustained Low Frequency Oscillation in Steady-State Power Systems

For sustained low frequency oscillations in steady-state power systems, an algorithm is proposed for precise online identification of oscillation frequency, oscillation amplitude, and fundamental amplitude. The algorithm consists of a robust low frequency estimator and a notch filter in parallel. The asymptotical convergence property is analyzed by slow integral manifold, averaging method, and Lyapunov stability theorem sequentially. The steady-state antinoise property is investigated by perturbed system theorem. The robust advantages of the proposed algorithm are embodied in the following aspects: the fundamental amplitude identification is little influenced by oscillation frequency and oscillation amplitude, both oscillation frequency identification and oscillation amplitude identification have small steady-state errors under high order harmonics or bounded noises, the ramp variations of both fundamental amplitude and oscillation amplitude are also significantly tracked, and three design parameters have different effects on identification performance of oscillation frequency, oscillation amplitude, and fundamental amplitude, respectively. Simulation results verify the validity.

A Toponogov-Type Comparison Theorem for Finsler Manifolds and its Applications

In this paper, we introduce the notion of modified flag curvature for Finsler metrics which naturally come from Hessian comparison theorem. We then establish a Toponogov-type comparison theorem for Finsler manifolds with nonnegative-modified flag curvature. As its applications, we prove that the fundamental group of any forward complete Finsler manifold (M, F) with nonnegative-modified flag curvature is finitely generated; furthermore, if additional M has bounded Cartan tensor, then M is of finite topological type.

On Kirchhoff Problems Involving Critical Exponent and Critical Growth

In this paper, we establish the existence of multiple solutions to a class of Kirchhoff type equations involving critical exponent, concave term and critical growth. Our main tools are the Nehari manifold and mountain pass theorem.

Multiple Solutions for an Elliptic Equation with Hardy-Sobolev Critical Exponent, Hardy-Sobolev-Maz’ya Potential and Sign-Changing Weights

In the present paper, an elliptic equation with Hardy-Sobolev critical exponent, Hardy-Sobolev-Maz’ya potential and sign-changing weights, is considered. By using the Nehari manifold and mountain pass theorem, the existence of at least four distinct solutions is obtained.

A generalized lower bound theorem for balanced manifolds

A simplicial complex of dimension $$d-1$$ is said to be balanced if its graph is d-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak Lefschetz property. As a corollary, we show that if $$\Delta $$ is an arbitrary balanced triangulation of any closed homology manifold of dimension $$d-1 \ge 3$$ , then $$2h_2(\Delta ) - (d-1)h_1(\Delta ) \ge 4{d \atopwithdelims ()2}(\tilde{\beta }_1(\Delta )-\tilde{\beta }_0(\Delta ))$$ , thus verifying a conjecture by Klee and Novik. To prove these results we develop the theory of flag $$h''$$ -vectors.

The Non-homogeneous Groshev Convergence theorem for Diophantine Approximation on Manifolds

This paper is based on Khintchine theorem, Groshev theorem and measure and dimension theorems for non-degenerate manifolds. The inhomogeneous Diophantine approximation of Groshev type on manifolds is studied. Major work is to discuss the inhomogeneous convergent theory of Diophantine approximation restricted to non-degenerate manifold in , based on the proof of Barker-Sprindzuk conjecture, the homogeneous theory of Diophantine approximation and inhomogeneous Groshev type theory for Diophantine approximation, by the decomposition of the set in manifold, with the aid of Borel Cantell lemma and transformation of lemma and its properties and the main inhomogeneous conversion principle, we know these two types of set in sense of Lebesgue measure is zero provided that the convergent sum condition is satisfied, from which several conclusions about the inhomogeneous convergent theory of Diophantine approximation is obtained. The main result is that Lebesgue measure is inhomogeneous strongly extremal. At last we use the fact that friendly measure is strongly contracting measure to develop an inhomogeneous strong extreme measure which is restricted to matrices with dependent quantities

Multiple Solutions to the Problem of Kirchhoff Type Involving the Critical Caffareli-Kohn-Niremberg Exponent, Concave Term and Sign-Changing Weights

In this paper, we establish the existence of at least four distinct solutions to an Kirchhoff type problems involving the critical Caffareli-Kohn-Niremberg exponent, concave term and sign-changing weights, by using the Nehari manifold and mountain pass theorem.

A classical approach to smooth supermanifolds

This is an extension of the author's Master's thesis written under the supervision of Dr. Gregor Weingart at the National Autonomous University of Mexico. By extension we mean that some of the results have been rewritten and some others have been added to the original work. The purpose of this study is to rewrite di erential supergeometry in terms of classical di erential geometry. This rewriting from rst principles has two main motivations: 1. avoid using local (and usually not very well-de ned) odd coordinates; 2. use both the language and the tools (both highly developed) of classical di erential geometry to state and prove results in supergeometry. Although there is work in this direction (for instance [SV86]) this work's point of view might be useful to translate from the sheaf-theoretic language to one that is better suited for explicit calculations. We now give a summary of the contents. Chapter 1 is about superalgebras. In particular, we give explicit isomorphisms for the space of superderivations of an exterior algebra. The nal section of this chapter is devoted to the twisted action of the symmetric group on the supervector space of tensors with a xed rank; this allows us to construct the supersymmetric and superexterior algebras of a nite-dimensional supervector space. In chapter 2 we begin our study of smooth nite-dimensional supermanifolds. As the title of this monograph indicates, our approach will be rather classical in the following sense: our de nition of a supermanifold is not given in terms of local charts nor in terms of sheaves of superalgebras; we rather study superalgebra bundles over smooth manifolds. That is, supermanifolds in our sense are vector bundles such that the bre at each point is a free supercommutative superalgebra of nite rank. This approach allows the use of tools from di erential geometry which in many cases (e.g. Batchelor's theorem, which we prove as corollary 2.12) simpli es the proofs; furthermore, we prove (theorem 2.11) that both approaches are equivalent. A noteworthy feature of our approach is that supersmooth maps turn out to be generalizations of linear di erential operators (proposition 2.6). With our approach we prove special splittings of the tangent bundle (theorem 2.25) and study the tangential maps of a supersmooth map between two supermanifolds which give conditions for the supermanifold to be split (proposition 2.36). We also prove that the de Rham cohomology of a supermanifold is isomorphic to the de Rham cohomology of the underlying smooth manifold (theorem 2.47) by writing down the exterior derivative in a di erent way (2.45). We include three appendices. The rst one concerns linear di erential operators; there we develop the necessary tools to understand supersmooth maps with our approach. The two other appendices deal with algebraic results of independent interest. The second appendix is the statement and proof of the Cartan Poincare lemma, a result on the (co)homology of a complex that arises in our study. The third appendix deals with the proof of an algebraic fact (lemma 1.26) we use in order to prove a owbox coordinates theorem for supermanifolds (theorem 2.34) which is the foundation of the main results of chapter 2.

Positive mass theorem for some asymptotically hyperbolic manifolds

We prove a positive mass theorem for some noncompact spin manifolds that are asymptotic to products of hyperbolic space with a compact manifold. As a conclusion we show the Yamabe inequality for some noncompact manifolds which are important to understand the behaviour of Yamabe invariants under surgeries.

The Gauss-Bonnet theorem

In this note we give a proof of the Gauss-Bonnet theorem for Riemannian manifolds (of any dimension) using Morse theory.

A Kastler–Kalau–Walze type theorem for five-dimensional manifolds with boundary

The Kastler–Kalau–Walze theorem, announced by A. Connes, shows that the Wodzicki residue of the inverse square of the Dirac operator is proportional to the Einstein–Hilbert action of general relativity. In this paper, we prove a Kastler–Kalau–Walze type theorem for five-dimensional manifolds with boundary.

Four Nontrivial Solutions for Kirchhoff Problems with Critical Potential, Critical Exponent and a Concave Term

In this paper, we consider the existence of multiple solutions to the Kirchhoff problems with critical potential, critical exponent and a concave term. Our main tools are the Nehari manifold and mountain pass theorem.