On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and boundary of the manifold. The principal angles between these interior subspaces are all acute and are called Poincare duality angles. This paper determines the Poincare duality angles of a collection of interesting manifolds with boundary derived from complex projective spaces and from Grassmannians, providing evidence that the Poincare duality angles measure, in some sense, how "close" a manifold is to being closed.
This paper also elucidates a connection between the Poincare duality angles and the Dirichlet-to-Neumann operator for differential forms, which generalizes the classical Dirichlet-to-Neumann map arising in the problem of Electrical Impedance Tomography. Specifically, the Poincare duality angles are essentially the eigenvalues of a related operator, the Hilbert transform for differential forms. This connection is then exploited to partially resolve a question of Belishev and Sharafutdinov about whether the Dirichlet-to-Neumann map determines the cup product structure on a manifold with boundary.
We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green's function vanishing at infinity. On the source function we assume a sharp pointwise decay depending on the weight appearing in the Poincaré inequality and on the behavior of the Ricci curvature at infinity. We do not require any curvature or spectral assumptions on the manifold.
We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary.
In this paper we prove the Poincaré lemma on some $n$-dimensional corank 1 sub-Riemannian structures, formulating the $\frac{(n-1)n(n^2+3n-2)}{8}$ necessarily and sufficiently 'curl-vanishing' compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. Our proof is based on a Poincaré lemma stated on Riemannian manifolds and a suitable Cesàro-Volterra path integral formula established in local coordinates. As a byproduct, a Saint-Venant lemma is also provided on generic Riemannian manifolds. Some examples are presented on the hyperbolic …
We prove the following version of Poincare duality for reduced $L_{q,p}$-cohomology: For any $1<q,p<\infty$, the $L_{q,p}$-cohomology of a Riemannian manifold is in duality with the interior $L_{p',q'}-cohomology for $1/p+1/p'=1$, $1/q+1/q'=1$.
We prove that the basic intersection cohomology $IH^*_{\overline{p}}(M / \mathcal{F})$, where $\mathcal{F}$ is the singular foliation determined by an isometric action of a Lie group $G$ on the compact manifold $M$, verifies the Poincaré Duality Property.
Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in \cite{BG}, then the following properties hold: 1 The volume doubling property; 2 The Poincaré inequality; 3 The parabolic Harnack inequality. The key ingredient is the study of dimensional reverse log-Sobolev inequalities for the heat semi…
Using a new type of Jacobi field estimate we will prove a duality theorem for singular Riemannian foliations in complete manifolds of nonnegative sectional curvature.
In the first part of this paper we prove some new Poincaré inequalities, with explicit constants, for domains of any hypersurface of a Riemannian manifold with sectional curvatures bounded from above. This inequalities involve the first and the second symmetric functions of the eigenvalues of the second fundamental form of such hypersurface. We apply these inequalities to derive some isoperimetric inequalities and to estimate the volume of domains enclosed by compact self-shrinkers in terms of its scalar curvature. In the second part of the paper we prove some mean value inequalities and as co…
Let $(M,g)$ be an open, oriented and incomplete riemannian manifold of dimension $m$. Under some general conditions we show that it is possible to build a Hilbert complex $(L^2Ω^i(M,g),d_{\mathfrak{M},i})$ such that its cohomology groups, labeled with $H^i_{2,\mathfrak{M}}(M,g)$, satisfy the following properties: \begin{itemize} \item $H^i_{2,\mathfrak{M}}(M,g)=ker(d_{max,i})/\im(d_{min,i})$ \item $H^i_{2,\mathfrak{M}}(M,g)\cong H^{m-i}_{2,\mathfrak{M}}(M,g)$ (Poincaré duality holds) \end{itemize} Finally in the rest of the paper we study some properties of this complex with particular attenti…
After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the Poincaré-Einstein structure, and obtain, from the renormalized integral of the Pfaffian, an extension of the Gauss-Bonnet theorem.
For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori's results, and an analogue of Petrie's conjecture. When G is an almost-connected Lie group or a discrete group, we establish Poincare duality between G-equivariant K-homology and K-theory, observing that Poincare duality does not necessarily hold for general G.
We provide an explicit description of the Poincaré dual of each generator of the rational cohomology ring of the $SU(2)$ character variety for a genus $g$ surface with central extension -- equivalently, that of the moduli space of stable holomorphic bundles of rank 2 and odd degree.
We revisit Spakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology. We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with uniform K-homology and prove Poincare duality between uniform K-theory and uniform K-homology on spin-c manifolds of bounded geometry.
We exhibit a Poisson module restoring a twisted Poincare duality between Poisson homology and cohomology for the polynomial algebra R=C[X_1,...,X_n] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier.
We study a Riemannian manifold equipped with a density which satisfies the Bakry--Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is locally-convex; this generalizes a purely Euclidean inequality of Colesanti, originally derived as an infinitesimal form of the Brunn-Minkowski inequality, thereby precluding any extensions beyond the Euclidean setting. A dual version for generalized mean-convex boundaries is also obt…
Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown, if the cone structure is regarded as a control system, then, the space of abnormal geodesics of the cone structure is naturally identified with the original space. In this paper, we provide an exposition on the duality by abnormal geodesics in a wider framework, namely, in terms of quotients of control systems and sub-Riemannian pseudo-product structures. Also we consider the controllability of cone structures and describe the co…
We carry out calculations of Orlicz cohomology for some basic Riemannian manifolds (the real line, the hyperbolic plane, the ball). Relationship between Orlicz cohomology and Poincaré--Sobolev--Orlicz-type inequalities is discussed.
Let $(M,g)$ an open and oriented riemannian manifold. The aim of this paper is to study some properties of the two following sequences of $L^2$ cohomology groups: $H^i_{2,m\rightarrow M}(M,g)$ defined as the image $\im(H^i_{2,min}(M,g)\rightarrow H^i_{2,max}(M,g))$ and $\bar{H}^i_{2,m\rightarrow M}(M,g)$ defined as $\im(\bar{H}^i_{2,min}(M,g)\rightarrow \bar{H}^i_{2,max}(M,g))$. We show, under certain hypothesis, that the first sequence is the cohomology of a suitable Hilbert complex which contains the minimal one and is contained in the maximal one. We also show that when the second sequence …
We study the spectral geometry of the conformal Jacobi operator on a 4-dimensional Riemannian manifold (M,g). We show that (M,g) is conformally Osserman if and only if (M,g) is self-dual or anti self-dual. Equivalently, this means that the curvature tensor of (M,g) is given by a quaternionic structure, at least pointwise.
For a Morse function on a closed orientable Riemannian manifold one introduces the {\it virtually small spectral package} an analytic object consisting of a finite number of analytic quantities derived from the pair, {\it Riemannian metric, Morse function\} which, in principle, can be calculated. One shows that they determine the {\it Torsion } of the underlying space, a parallel to the result that the dimensions of the spaces of harmonic forms calculate the {\it Euler-Poincaré characteristic} of the underlying space and extends the {\it Poincaré Duality} between harmonic forms and between Bet…
We prove a Poincaré, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature bounded above. Our techniques allow us to consider Riemannian manifolds $(M^n,g)$ with $g$ of class $C^2$ or more regular, avoiding the use of Nash's isometric embedding theorem. Our analysis permits to do some quite important fragments of geometric measure theory also for those Riemannian manifolds carrying a $C^2$ metric $g$, that is not $C^{k+α}$ with $k+α>2…
In those lecture notes, we review some applications of heat semigroups methods in Riemannian and sub-Riemannian geometry. The notes contain parts of courses taught at Purdue University, Institut Henri Poincaré, Levico Summer School and Tata Institute.
An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are special cases. The governing equation is a conformally invariant overdetermined PDE on a function. Away from the zeros of this the almost Einstein structure is Einstein, while the zero set gives a scale singularity set which may be viewed as a conformal infinity for the Einstein metric. In this article we give a classification of the possible scale singular…
Given a closed Riemannian manifold $(M, g_M)$ of dimension $n \geq 3$, we prove the existence of a conformally compact Einstein metric $g_{+}$ defined on a collar neighborhood $M \times (0,1]$ whose conformal infinity is $[g_M]$.
We study the validity of the $L^p$ inequality for the Riesz transform when $p>2$ and of its reverse inequality when $p<2$ on complete Riemannian manifolds under the doubling property and some Poincaré inequalities.
We study the cohomology properties of the singular foliation $\F$ determined by an action $Φ\colon G \times M\to M$ where the abelian Lie group $G$ preserves a riemannian metric on the compact manifold $M$. More precisely, we prove that the basic intersection cohomology $\lau{\IH}{*}{\per{p}}{\mf}$ is finite dimensional and verifies the Poincaré Duality. This duality includes two well-known situations: -- Poincaré Duality for basic cohomology (the action $Φ$ is almost free). -- Poincaré Duality for intersection cohomology (the group $G$ is compact and connected).
By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and N. Garofalo in \cite{Bau2}. Our results apply in particular on all CR Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is non negative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is …
Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）；Ludwig Maximilian University（慕尼黑大学）；Universitá degli Studi di Bari ‘Aldo Moro’（巴里阿尔多·莫罗大学）
The more important difference between Riemann and pseudo-Riemann manifolds is the metric signature and its theoretical consequences. The practical application for Physics Theories becomes often impossible due to the signature consequences. Eg., some of the rich results in Riemann Geometry and Topology become invalid for Physics if they are based on the concept of the positive definite norm; to avoid this problem, the proof machinery must avoid such assumption and must be based in other tools. This paper is a contribution to provide methodologies for Hodge decomposition and \poincare duality ba…
We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincaré-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorphic family of distributions with residues given by the residue family operators $\slashed{D}_N^{res}(h;λ)$ on spinors. We develop basic aspects and properties of $\slashed{D}_N^{res}(h;λ)$ including conformal covariance, factorization properties by conformally covariant operators…
