Pub Date : 2020-01-24DOI: 10.1142/S0219887820501224
A. Bruce
We show how to lift a Riemannian metric and almost symplectic form on a manifold to a Riemannian structure on a canonically associated supermanifold known as the antitangent or shifted tangent bundle. We view this construction as a generalisation of Sasaki's construction of a Riemannian metric on the tangent bundle of a Riemannian manifold.
{"title":"The super-Sasaki metric on the antitangent bundle","authors":"A. Bruce","doi":"10.1142/S0219887820501224","DOIUrl":"https://doi.org/10.1142/S0219887820501224","url":null,"abstract":"We show how to lift a Riemannian metric and almost symplectic form on a manifold to a Riemannian structure on a canonically associated supermanifold known as the antitangent or shifted tangent bundle. We view this construction as a generalisation of Sasaki's construction of a Riemannian metric on the tangent bundle of a Riemannian manifold.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78700466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In this paper we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data.
{"title":"Expected distances on manifolds of partially oriented flags","authors":"Brenden Balch, C. Peterson, C. Shonkwiler","doi":"10.1090/proc/15521","DOIUrl":"https://doi.org/10.1090/proc/15521","url":null,"abstract":"Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In this paper we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84633937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-14DOI: 10.1016/J.MATPUR.2020.09.001
A. Latorre, L. Ugarte
{"title":"On the stability of compact pseudo-Kähler and neutral Calabi-Yau manifolds","authors":"A. Latorre, L. Ugarte","doi":"10.1016/J.MATPUR.2020.09.001","DOIUrl":"https://doi.org/10.1016/J.MATPUR.2020.09.001","url":null,"abstract":"","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"92 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83777647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss the decomposition of degree two basic cohomology for codimension four taut Riemannian foliation according to the holonomy invariant transversal almost complex structure J, and show that J is C pure and full. In addition, we obtain an estimate of the dimension of basic J-anti-invariant subgroup. These are the foliated version for the corresponding results of T. Draghici et al.
{"title":"Direct sum for basic cohomology of codimension four taut Riemannian foliation","authors":"Jiuru Zhou","doi":"10.4134/BKMS.B200022","DOIUrl":"https://doi.org/10.4134/BKMS.B200022","url":null,"abstract":"We discuss the decomposition of degree two basic cohomology for codimension four taut Riemannian foliation according to the holonomy invariant transversal almost complex structure J, and show that J is C pure and full. In addition, we obtain an estimate of the dimension of basic J-anti-invariant subgroup. These are the foliated version for the corresponding results of T. Draghici et al.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72910835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We estimate from above the rate at which a solution to the normalized Ricci flow on a closed manifold may converge to a limit soliton. Our main result implies that any solution which converges modulo diffeomorphisms to a soliton faster than any fixed exponential rate must itself be self-similar.
{"title":"On the Maximal Rate of Convergence Under the Ricci Flow","authors":"Brett L. Kotschwar","doi":"10.1093/imrn/rnaa172","DOIUrl":"https://doi.org/10.1093/imrn/rnaa172","url":null,"abstract":"We estimate from above the rate at which a solution to the normalized Ricci flow on a closed manifold may converge to a limit soliton. Our main result implies that any solution which converges modulo diffeomorphisms to a soliton faster than any fixed exponential rate must itself be self-similar.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74277215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with deformations of Kundt metrics in the direction of type $III$ tensors and nil-Killing vector fields whose flows give rise to such deformations. We find various characterizations within the Kundt class in terms of nil-Killing vector fields and obtain a theorem classifying algebraic stability of tensors, which has an application in finding sufficient criteria for a type $III$ deformation of the metric to preserve spi's. This is used in order to specify Lie algebras of nil-Killing vector fields that preserve the spi's, for degenerate Kundt metrics. Using this we discuss the characterization of Kundt-CSI spacetimes in terms of nil-Killing vector fields.
{"title":"Nil-Killing vector fields and type III deformations","authors":"M. Aadne","doi":"10.1063/5.0018773","DOIUrl":"https://doi.org/10.1063/5.0018773","url":null,"abstract":"This paper is concerned with deformations of Kundt metrics in the direction of type $III$ tensors and nil-Killing vector fields whose flows give rise to such deformations. We find various characterizations within the Kundt class in terms of nil-Killing vector fields and obtain a theorem classifying algebraic stability of tensors, which has an application in finding sufficient criteria for a type $III$ deformation of the metric to preserve spi's. This is used in order to specify Lie algebras of nil-Killing vector fields that preserve the spi's, for degenerate Kundt metrics. Using this we discuss the characterization of Kundt-CSI spacetimes in terms of nil-Killing vector fields.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89435643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-01DOI: 10.5666/KMJ.2020.60.1.185
H. Abu-Donia, S. Shenawy, A. Syied
This paper aims to study the $W$-curvature tensor on relativistic space-times. The energy-momentum tensor T of a space-time is semi-symmetric given that the $W$-curvature tensor is semi-symmetric whereas energy-momentum tensor T of a space-time having a divergence free $W$-curvature tensor is of Codazzi type. A space-time having a traceless $W$-curvature tensor is Einstein. A $W$-curvature flat space-time is Einstein. Perfect fluid space-times which admits $W$-curvature tensor are considered.
{"title":"The $W$-curvature tensor on relativistic space-times","authors":"H. Abu-Donia, S. Shenawy, A. Syied","doi":"10.5666/KMJ.2020.60.1.185","DOIUrl":"https://doi.org/10.5666/KMJ.2020.60.1.185","url":null,"abstract":"This paper aims to study the $W$-curvature tensor on relativistic space-times. The energy-momentum tensor T of a space-time is semi-symmetric given that the $W$-curvature tensor is semi-symmetric whereas energy-momentum tensor T of a space-time having a divergence free $W$-curvature tensor is of Codazzi type. A space-time having a traceless $W$-curvature tensor is Einstein. A $W$-curvature flat space-time is Einstein. Perfect fluid space-times which admits $W$-curvature tensor are considered.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"99 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75931243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the oscillator group ${rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of ${rm Osc}_1$ up to inner automorphisms of ${rm Osc}_1$. For every lattice $L$ in ${rm Osc}_1$, we compute the decomposition of the right regular representation of ${rm Osc}_1$ on $L^2(Lbackslash{rm Osc}_1)$ into irreducible unitary representations. This decomposition is called the spectrum of the quotient $Lbackslash{rm Osc}_1$.
{"title":"Spectra of compact quotients of the oscillator group","authors":"Mathias Fischer, I. Kath","doi":"10.3842/SIGMA.2021.051","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.051","url":null,"abstract":"We consider the oscillator group ${rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of ${rm Osc}_1$ up to inner automorphisms of ${rm Osc}_1$. For every lattice $L$ in ${rm Osc}_1$, we compute the decomposition of the right regular representation of ${rm Osc}_1$ on $L^2(Lbackslash{rm Osc}_1)$ into irreducible unitary representations. This decomposition is called the spectrum of the quotient $Lbackslash{rm Osc}_1$.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86031421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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