A note on tensor fields in Hilbert spaces

Biliotti, Leonardo; Mercuri, Francesco; Tausk, Daniel V.

doi:10.1590/s0001-37652002000200003

Cited by 4 publications

(3 citation statements)

References 2 publications

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“…The aim of this survey is to describe results obtained by the authors jointly with D. Tausk, R. Exel and P. Piccione and others authors [1,2,3,4,5,6,7,11,13,24]. We have tried to avoid technical results in order to make the paper more readable also by non experts in this field.…”

Section: Introductionmentioning

confidence: 99%

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Riemannian Hilbert Manifolds

Biliotti

Mercuri

2017

Springer Proceedings in Mathematics &Amp; Statistics

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In this article we collect results obtained by the authors jointly with other authors and we discuss old and new ideas. In particular we discuss singularities of the exponential map, completeness and homogeneity for Riemannian Hilbert quotient manifolds. We also extend a Theorem due to Nomizu and Ozeki to infinite dimensional Riemannian Hilbert manifolds.2000 Mathematics Subject Classification. 58B99;57S25.

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Section: Introductionmentioning

confidence: 99%

“…i e i ) = x 1 e 1 + x 2 e 2 + ∞ i=3 (−x i )e i .For any k ≥ 3, E k := {x2 1 ֒→ M is a totally geodesic submanifold of M since it is the fixed points set of the isometryF ( ∞ i=1 x i e i ) = x 1 e 1 − x 2 e 2 + x k e k + ∞ i=3,i =k (−x i )e i . Hence K( γ(s), e k ) = (1− 1 k ) 2 , J k (t) = sin(t(1− 1 k…”

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confidence: 99%

Riemannian Hilbert Manifolds

Biliotti

Mercuri

2017

Springer Proceedings in Mathematics &Amp; Statistics

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“…Demonstração: sejam C' Yl, --y 2 , --y 3 ) e (""1 1 , --y 2 , ~r 3 ) dois triângulos em MH-' com li= Ii, para i=1,2,3. Pelo passo (1) --y 3 é univocamente determinada por a 3 ; então a 3 = a 3 e por isso a isometria que manda ~li em 'fi, i=1,2, leva (""1 1 , ~(2, ""/3) em (--y,, ""12, ""/3)• (5) que (ak, Tk,l+l, ak) verifica (B). (11) Assumimos N como no item (9) Seja rEM tal que d(p, r) 2' : 7r -E. Aplicando o corolário (6.2.5) do teorema de Topogonov ao triângulo geodésico de vértices (p, q, r) temos Agora, a prova continua exatamente como no teorema da Esfera (Rauch).…”

Section: Teorema De Topogonovunclassified

Alguns aspectos da geometria riemanniana das variedades de Hilbert

Biliotti¹,

Mercuri²

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The aim of this work is to formalize the local theory of infinite dimensional Riemannian manifold and to study the geometry /topology when the sectional curvature is bounded by two positive constant. We compare this situation with the finite dimensional case and emphasize the difference. The local theory was already developed since 1960, so we describe, briefly, the basic facts of the theory as the existence and uniqueness of Levi Civita connection, Gauss lemma and existence of convex neighborhood. However, we proved that the fundamental theorem of tensor field is not verified and we introduced a class, that we called C 00 -weakly ,for which the criterion holds. When we want to study the global properties, the fact that the manifold is complete is fundamental, as in finite dimensional case, but as the Hopf-Rinow theorem is not always verified, completeness is not always equivalent to geodesic completeness. These pathology is not verified by complete simply connected manifolds with constant sectional curvature and we have the same classification as the finite dimensional case. However, the class o f infinite dimensional manifolds of constant positive curvatura is bigger than the respective class in the finite dimensional case. This fact is consequence of the study of the groups that could acts effective and properly discontinuosly, as isometry group, on the unitary sphere on infinite dimensional Hilbert spaces. The two basics facts that justify our are that any infinite dimensional Hilbert space and any group G without torsion, acts without fixed point and properly discontinuous, as isometry group, on the sphere in 1 2 ( G). The extension o f theorems of Rauch and Topogonov, is fundamental when we study the geometry of with sectional curvature bounded by two positive constants. The consequence are the extension of some classical result, that we have proved in chapter 6, like of Berger-Topogonov theorem about the maximal diameter and, when we assume that the radius of injectivity of the manifolds is at least 'IT, some results in the spirit of the pinching theorems. !V Vll Vlll AgradecimentosAlia mia famiglia, babbo mamma e fratello, per avermi insegnato a lottare per ottenere i risultati desiderati, per avermi dato la possibilità di studiare, sempre nella massima libertà, e per avermi sopportato nei momenti in cui il mio nervosismo si rifletteva nel mio relazionemento con !oro.A Selma, minha namorada e companheira destes últimos dois anos, pelo amor, os carinhos, mas principalmente pelo fato de existir e de me ter escolhido como seu namorado.Ao Conselho Nacional de Pesquisa (CNPq) pelo suporte financeiro.Ringrazio tantissimo il mio "Orientador" Francesco Mercuri per avermi dato la possibilità di studiare all' Unicamp, per tutti i suoi innumerevoli consigli e per la disponibilità offerta anche al di fuori del normale relazionamento di alunno e professore.Ao meu Co-Orientador Daniel Victor Tausk pela disponibilidade oferecida toda vez que pedi sua ajuda.Ao Prof. Adonai pelas ótimas conversas matemáticas e pelas ajuda...

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Properly discontinuous actions on Hilbert manifolds

Biliotti

Mercuri

2014

Bull Braz Math Soc, New Series

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In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrast with the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant sectional curvature.2000 Mathematics Subject Classification. 58B99;57S25.

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A note on tensor fields in Hilbert spaces

Abstract: We discuss and extend to infinite dimensional Hilbert spaces a well-known tensoriality criterion for linear endomorphisms of the space of smooth vector fields in R n .

Cited by 4 publications

References 2 publications

Riemannian Hilbert Manifolds

Riemannian Hilbert Manifolds

Alguns aspectos da geometria riemanniana das variedades de Hilbert

Properly discontinuous actions on Hilbert manifolds

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