scholarly journals Riesz transforms of the Hodge-de Rham Laplacian on Riemannian manifolds

2015 ◽  
Vol 289 (8-9) ◽  
pp. 1021-1043 ◽  
Author(s):  
Jocelyn Magniez
Keyword(s):  
Riemannian Manifolds ◽  
Riesz Transforms ◽  
De Rham

scholarly journals The Hodge–de Rham Laplacian and Lp-boundedness of Riesz transforms on non-compact manifolds

2015 ◽  
Vol 125 ◽  
pp. 78-98 ◽  
Author(s):  
Peng Chen ◽  
Jocelyn Magniez ◽  
El Maati Ouhabaz
Keyword(s):  
Riesz Transforms ◽  
Compact Manifolds ◽  
De Rham

scholarly journals Chiral de Rham Complex on Riemannian Manifolds and Special Holonomy

2013 ◽  
Vol 318 (3) ◽  
pp. 575-613 ◽  
Author(s):  
Joel Ekstrand ◽  
Reimundo Heluani ◽  
Johan Källén ◽  
Maxim Zabzine
Keyword(s):  
Riemannian Manifolds ◽  
De Rham Complex ◽  
Special Holonomy ◽  
Rham Complex ◽  
De Rham

scholarly journals A Decomposition Theorem for Immersions of Product Manifolds

2015 ◽  
Vol 59 (1) ◽  
pp. 247-269 ◽  
Author(s):  
Ruy Tojeiro
Keyword(s):  
Riemannian Manifolds ◽  
Space Form ◽  
Type Theorem ◽  
Decomposition Theorem ◽  
Product Manifold ◽  
Isometric Immersions ◽  
Product Metrics ◽  
Special Cases ◽  
De Rham ◽  
Shape Operators

AbstractWe introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally or globally decomposed as a product manifold endowed with a polar metric. For such a product manifold, our main result gives a complete description of all its isometric immersions into a space form whose second fundamental forms are adapted to its product structure in the sense that the tangent spaces to each factor are preserved by all shape operators. This is a far-reaching generalization of a basic decomposition theorem for isometric immersions of Riemannian products due to Moore as well as of its extension by Nölker to isometric immersions of warped products.

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

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Marek Grochowski
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Type Theorem ◽  
Parallel Transport ◽  
Holonomy Group ◽  
Decomposition Theorem ◽  
Reeb Vector Field ◽  
Indefinite Metrics ◽  
De Rham ◽  
Riemannian Case

AbstractIn 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 $$\xi $$ ξ is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point $$q\in M$$ q ∈ M such that the holonomy group $$\Psi (q)$$ Ψ ( q ) acts reducibly on H(q) yielding a decomposition $$H(q) = H_1(q)\oplus \cdots \oplus H_m(q)$$ H ( q ) = H 1 ( q ) ⊕ ⋯ ⊕ H m ( q ) into $$\Psi (q)$$ Ψ ( q ) -irreducible factors. Using parallel transport we obtain the decomposition $$H = H_1\oplus \cdots \oplus H_m$$ H = H 1 ⊕ ⋯ ⊕ H m of H into sub-distributions $$H_i$$ H i . Unlike the Riemannian case, the distributions $$H_i$$ H i are not integrable, however they induce integrable distributions $$\Delta _i$$ Δ i on $$M/\xi $$ M / ξ , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that $$T(U/\xi )=\Delta _1\oplus \cdots \oplus \Delta _m$$ T ( U / ξ ) = Δ 1 ⊕ ⋯ ⊕ Δ m , and the latter decomposition of $$T(U/\xi )$$ T ( U / ξ ) induces the decomposition of $$U/\xi $$ U / ξ 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.

scholarly journals Some Algebraic and Geometric Structures on Poisson Manifolds

2005 ◽  
Vol 12 (1) ◽  
pp. 171-179
Author(s):  
Zaza Tevdoradze
Keyword(s):  
Riemannian Manifolds ◽  
Classical Case ◽  
Poisson Manifolds ◽  
Harmonic Form ◽  
Harmonic Forms ◽  
De Rham Cohomology ◽  
Geometric Structures ◽  
Algebraic Properties ◽  
De Rham

Abstract In this paper we study some algebraic properties of harmonic forms on Poisson manifolds. It is well known that in the classical case (on Riemannian manifolds) the product of harmonic forms is not harmonic. Here we describe the algebraic and analytical mechanisms explaining this fact. We also obtain a condition under which the product of de Rham cohomology classes, which includes harmonic representatives, can be represented by a harmonic form.

scholarly journals On the L2-Poincaré duality for incomplete Riemannian manifolds: A general construction with applications

2016 ◽  
Vol 08 (01) ◽  
pp. 151-186 ◽  
Author(s):  
Francesco Bei
Keyword(s):  
Riemannian Manifolds ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Finite Dimensional Vector Space ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Rham Complex ◽  
De Rham ◽  
Hilbert Complex ◽  
Incomplete Riemannian Manifolds

Let [Formula: see text] be an open, oriented and incomplete Riemannian manifold of dimension [Formula: see text]. Under some general conditions we show the existence of a Hilbert complex [Formula: see text] such that its cohomology groups, labeled with [Formula: see text], satisfy the following properties: [Formula: see text] [Formula: see text] (Poincaré duality holds) There exists a well-defined and nondegenerate pairing: [Formula: see text] If [Formula: see text] is a Fredholm complex, then every closed extension of the de Rham complex [Formula: see text] is a Fredholm complex and, for each [Formula: see text], the quotient [Formula: see text] is a finite dimensional vector space.

scholarly journals Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds

2011 ◽  
Vol 61 (4) ◽  
pp. 1323-1349 ◽  
Author(s):  
Steve Hofmann ◽  
Marius Mitrea ◽  
Sylvie Monniaux
Keyword(s):  
Riemannian Manifolds ◽  
Riesz Transforms ◽  
Hodge Laplacian
Sign in / Sign up

Export Citation Format

Share Document