De Rham-Hodge theory for Riemannian foliations

Kamber, Franz W.; Tondeur, Philippe

doi:10.1007/bf01458323

Cited by 93 publications

(62 citation statements)

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“…If rank(g f ) = constant then g f is a Riemannian foliation as in [38][39][40]46]. If rank(g f ) is not constant we apply [62].…”

Section: Discussionmentioning

confidence: 99%

“…For the general theory of Riemannian foliations the readers are referred to [39,40,46], see also the monograph [38] and the references therein.…”

Section: Definition 26mentioning

confidence: 99%

“…This subsection is mainly devoted to examples of stratified Riemannian foliations in analytic manifolds. For more details about those object the readers are referred to [46,63]. Theorem 24.…”

Section: Stratified Analytic Riemannian Foliationsmentioning

confidence: 99%

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Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Boyom

2016

Entropy

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Let us begin by considering two book titles: A provocative title, What Is a Statistical Model? McCullagh (2002) and an alternative title, In a Search for Structure. The Fisher Information. Gromov (2012). It is the richness in open problems and the links with other research domains that make a research topic exciting. Information geometry has both properties. Differential information geometry is the differential geometry of statistical models. The topology of information is the topology of statistical models. This highlights the importance of both questions raised by Peter McCullagh and Misha Gromov. The title of this paper looks like a list of key words. However, the aim is to emphasize the links between those topics. The theory of homology of Koszul-Vinberg algebroids and their modules (KV homology in short) is a useful key for exploring those links. In Part A we overview three constructions of the KV homology. The first construction is based on the pioneering brute formula of the coboundary operator. The second construction is based on the theory of semi-simplicial objects. The third construction is based on the anomaly functions of abstract algebras and their abstract modules. We use the KV homology for investigating links between differential information geometry and differential topology. For instance, "dualistic relation of Amari" and "Riemannian or symplectic Foliations"; "Koszul geometry" and "linearization of webs"; "KV homology" and "complexity of models". Regarding the complexity of a model, the challenge is to measure how far from being an exponential family is a given model. In Part A we deal with the classical theory of models. Part B is devoted to answering both questions raised by McCullagh and B Gromov. A few criticisms and examples are used to support our criticisms and to motivate a new approach. In a given category an outstanding challenge is to find an invariant which encodes the points of a moduli space. In Part B we face four challenges. (1) The introduction of a new theory of statistical models. This re-establishment must answer both questions of McCullagh and Gromov; (2) The search for an characteristic invariant which encodes the points of the moduli space of isomorphism class of models; (3) The introduction of the theory of homological statistical models. This is a pioneering notion. We address its links with Hessian geometry; (4) We emphasize the links between the classical theory of models, the new theory and Vanishing Theorems in the theory of homological statistical models. Subsequently, the differential information geometry has a homological nature. That is another notable feature of our approach. This paper is dedicated to our friend and colleague Alexander Grothendieck.

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“…If rank(g f ) = constant then g f is a Riemannian foliation as in [38][39][40]46]. If rank(g f ) is not constant we apply [62].…”

Section: Discussionmentioning

confidence: 99%

“…For the general theory of Riemannian foliations the readers are referred to [39,40,46], see also the monograph [38] and the references therein.…”

Section: Definition 26mentioning

confidence: 99%

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Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Boyom

2016

Entropy

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“…It is interesting to find situations where this operator will give generalizations to known theorems concerning the ordinary Laplacian. For examples of research in this area, see [1], [6], [7], [8], [11], [13], and [18].…”

Section: Introductionmentioning

confidence: 99%

Lichnerowicz and Obata theorems for foliations

Lee¹,

Richardson²

2002

Pacific J. Math.

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The standard Lichnerowicz comparison theorem states that if the Ricci curvature of a closed, Riemannian n-manifold M satisfies Ric (X, X) ≥ a (n − 1) |X| 2 for every X ∈ T M for some fixed a > 0, then the smallest positive eigenvalue λ of the Laplacian satisfies λ ≥ an. The Obata theorem states that equality occurs if and only if M is isometric to the standard n-sphere of constant sectional curvature a. In this paper, we prove that if M is a closed Riemannian manifold with a Riemannian foliation of codimension q, and if the normal Ricci curvature satisfies Ric ⊥ (X, X) ≥ a (q − 1) |X| 2 for every X in the normal bundle for some fixed a > 0, then the smallest eigenvalue λ B of the basic Laplacian satisfies λ B ≥ aq. In addition, if equality occurs, then the leaf space is isometric to the space of orbits of a discrete subgroup of O (q) acting on the standard q-sphere of constant sectional curvature a. We also prove a result about bundle-like metrics on foliations: On any Riemannian foliation with bundle-like metric, there exists a bundle-like metric for which the mean curvature is basic and the basic Laplacian for the new metric is the same as that of the original metric.

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“…Now we define the connection ∇ on Ω * B (F), which is induced from the connection ∇ on Q and Riemannian connection ∇ M of g M . This connection ∇ extends the partial Bott connection [10]. Then the basic forms are characterized by Ω * B (F) = Ker…”

Section: Preliminariesmentioning

confidence: 99%