Pub Date : 2020-07-27DOI: 10.31926/but.mif.2020.13.62.1.11
K. Eftekharinasab, V. Petrusenko
We establish a framework, namely, nuclear bounded Frechet manifolds endowed with Riemann-Finsler structures to study geodesic curves on certain infinite dimensional manifolds such as the manifold of Riemannian metrics on a closed manifold. We prove on these manifolds geodesics exist locally and they are length minimizing in a sense. Moreover, we show that a curve on these manifolds is geodesic if and only if it satisfies a collection of Euler-Lagrange equations. As an application, without much difficulty, we prove that the solution to the Ricci flow on an Einstein manifold is not geodesic.
{"title":"Finslerian geodesics on Frechet manifolds","authors":"K. Eftekharinasab, V. Petrusenko","doi":"10.31926/but.mif.2020.13.62.1.11","DOIUrl":"https://doi.org/10.31926/but.mif.2020.13.62.1.11","url":null,"abstract":"We establish a framework, namely, nuclear bounded Frechet manifolds endowed with Riemann-Finsler structures to study geodesic curves on certain infinite dimensional manifolds such as the manifold of Riemannian metrics on a closed manifold. We prove on these manifolds geodesics exist locally and they are length minimizing in a sense. Moreover, we show that a curve on these manifolds is geodesic if and only if it satisfies a collection of Euler-Lagrange equations. As an application, without much difficulty, we prove that the solution to the Ricci flow on an Einstein manifold is not geodesic.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82964923","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}
The intent of this short note is to extend real valued Lipschitz functions on metric spaces, while locally preserving the asymptotic Lipschitz constant. We then apply this results to give a simple and direct proof of the fact that Sobolev spaces on metric measure spaces defined with a relaxation approach a la Cheeger are invariant under isomorphism class of mm-structures.
{"title":"Global Lipschitz extension preserving local constants","authors":"Simone Di Marino, N. Gigli, A. Pratelli","doi":"10.4171/RLM/913","DOIUrl":"https://doi.org/10.4171/RLM/913","url":null,"abstract":"The intent of this short note is to extend real valued Lipschitz functions on metric spaces, while locally preserving the asymptotic Lipschitz constant. We then apply this results to give a simple and direct proof of the fact that Sobolev spaces on metric measure spaces defined with a relaxation approach a la Cheeger are invariant under isomorphism class of mm-structures.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"199 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72923327","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}
Let $B^nsubset {mathbb R}^{n}$ and ${mathbb S}^nsubset {mathbb R}^{n+1}$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. The textit{extrinsic $k$-energy functional} is defined on the Sobolev space $W^{k,2}left (B^n,{mathbb S}^n right )$ as follows: $E_{k}^{rm ext}(u)=int_{B^n}|Delta^s u|^2,dx$ when $k=2s$, and $E_{k}^{rm ext}(u)=int_{B^n}|nabla Delta^s u|^2,dx$ when $k=2s+1$. These energy functionals are a natural higher order version of the classical extrinsic bienergy, also called Hessian energy. The equator map $u^*: B^n to {mathbb S}^n$, defined by $u^*(x)=(x/|x|,0)$, is a critical point of $E_{k}^{rm ext}(u)$ provided that $n geq 2k+1$. The main aim of this paper is to establish necessary and sufficient conditions on $k$ and $n$ under which $u^*: B^n to {mathbb S}^n$ is minimizing or unstable for the extrinsic $k$-energy.
设$B^nsubset {mathbb R}^{n}$和${mathbb S}^nsubset {mathbb R}^{n+1}$分别表示欧几里得$n$维单位球和球。textit{外在的$k$-能量泛函}在Sobolev空间$W^{k,2}left (B^n,{mathbb S}^n right )$上定义如下:$k=2s$时为$E_{k}^{rm ext}(u)=int_{B^n}|Delta^s u|^2,dx$, $k=2s+1$时为$E_{k}^{rm ext}(u)=int_{B^n}|nabla Delta^s u|^2,dx$。这些能量泛函是经典外在生物能量的自然高阶版本，也称为黑森能量。由$u^*(x)=(x/|x|,0)$定义的赤道图$u^*: B^n to {mathbb S}^n$是$E_{k}^{rm ext}(u)$的一个临界点，假设$n geq 2k+1$。本文的主要目的是建立$k$和$n$的充分必要条件，在此条件下，$u^*: B^n to {mathbb S}^n$对于外部的$k$ -能量是最小的或不稳定的。
{"title":"On the Stability of the Equator Map for Higher Order Energy Functionals","authors":"A. Fardoun, S. Montaldo, A. Ratto","doi":"10.1093/IMRN/RNAB009","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB009","url":null,"abstract":"Let $B^nsubset {mathbb R}^{n}$ and ${mathbb S}^nsubset {mathbb R}^{n+1}$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. The textit{extrinsic $k$-energy functional} is defined on the Sobolev space $W^{k,2}left (B^n,{mathbb S}^n right )$ as follows: $E_{k}^{rm ext}(u)=int_{B^n}|Delta^s u|^2,dx$ when $k=2s$, and $E_{k}^{rm ext}(u)=int_{B^n}|nabla Delta^s u|^2,dx$ when $k=2s+1$. These energy functionals are a natural higher order version of the classical extrinsic bienergy, also called Hessian energy. The equator map $u^*: B^n to {mathbb S}^n$, defined by $u^*(x)=(x/|x|,0)$, is a critical point of $E_{k}^{rm ext}(u)$ provided that $n geq 2k+1$. The main aim of this paper is to establish necessary and sufficient conditions on $k$ and $n$ under which $u^*: B^n to {mathbb S}^n$ is minimizing or unstable for the extrinsic $k$-energy.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80107793","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 : 2020-07-01DOI: 10.15673/TMGC.V14I1.1784
Jean-Pierre Magnot
In this article, we examine the geometry of a group of Fourier-integral operators, which is the central extension of Dif f (S 1) with a group of classical pseudo-differential operators of any order. Several subgroups are considered , and the corresponding groups with formal pseudodifferential operators are defined. We investigate the relationship of this group with the restricted general linear group GLres, we define a right-invariant pseudo-Riemannian metric on it that extends the Hilbert-Schmidt Riemannian metric by the use of renormalized traces of pseudo-differential operators, and we describe classes of remarkable connections. MSC (2010) : 22E66, 47G30, 58B20, 58J40
{"title":"On the geometry of $Diff(S^1)$-pseudodifferential operators based on renormalized traces.","authors":"Jean-Pierre Magnot","doi":"10.15673/TMGC.V14I1.1784","DOIUrl":"https://doi.org/10.15673/TMGC.V14I1.1784","url":null,"abstract":"In this article, we examine the geometry of a group of Fourier-integral operators, which is the central extension of Dif f (S 1) with a group of classical pseudo-differential operators of any order. Several subgroups are considered , and the corresponding groups with formal pseudodifferential operators are defined. We investigate the relationship of this group with the restricted general linear group GLres, we define a right-invariant pseudo-Riemannian metric on it that extends the Hilbert-Schmidt Riemannian metric by the use of renormalized traces of pseudo-differential operators, and we describe classes of remarkable connections. MSC (2010) : 22E66, 47G30, 58B20, 58J40","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"362 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91473122","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 : 2020-07-01DOI: 10.1007/978-3-030-58653-9_8
M. G. Molina, I. Markina
{"title":"Sub-Riemannian Geodesics on Nested Principal Bundles","authors":"M. G. Molina, I. Markina","doi":"10.1007/978-3-030-58653-9_8","DOIUrl":"https://doi.org/10.1007/978-3-030-58653-9_8","url":null,"abstract":"","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73144156","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 define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are shown to simultaneously generalise regularly foliated bundles in the sense of Kamber-Tondeur, bundles that are equivariant under the actions Lie groupoids with simply connected source fibres, and singular foliations. We define hierarchies of diffeological holonomy groupoids associated to such bundles, which arise from the parallel transport of germs of local conservation laws on the base that take values in the total space. In particular, for any singular foliation with "enough" local conservation laws, our construction recovers the holonomy groupoid defined by Androulidakis and Skandalis as a special case. Finally we prove functoriality of all our constructions under appropriate morphisms.
{"title":"The Holonomy Groupoids of Singularly Foliated Bundles","authors":"L. MacDonald","doi":"10.3842/SIGMA.2021.043","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.043","url":null,"abstract":"We define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are shown to simultaneously generalise regularly foliated bundles in the sense of Kamber-Tondeur, bundles that are equivariant under the actions Lie groupoids with simply connected source fibres, and singular foliations. We define hierarchies of diffeological holonomy groupoids associated to such bundles, which arise from the parallel transport of germs of local conservation laws on the base that take values in the total space. In particular, for any singular foliation with \"enough\" local conservation laws, our construction recovers the holonomy groupoid defined by Androulidakis and Skandalis as a special case. Finally we prove functoriality of all our constructions under appropriate morphisms.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"199 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77982441","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 : 2020-06-25DOI: 10.1007/978-3-030-63403-2_2
Hans-Peter Schrocker, M. Pfurner, Johannes Siegele
{"title":"Space Kinematics and Projective Differential Geometry over the Ring of Dual Numbers","authors":"Hans-Peter Schrocker, M. Pfurner, Johannes Siegele","doi":"10.1007/978-3-030-63403-2_2","DOIUrl":"https://doi.org/10.1007/978-3-030-63403-2_2","url":null,"abstract":"","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"3 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77499022","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 give a one-parameter family of examples of shrinking Laplacian solitons, which are the second known solutions to the closed $G_2$-Laplacian flow with a finite-time singularity. The torsion forms and the Laplacian and Ricci operators of a large family of $G_2$-structures on different Lie groups are also studied. We apply these formulas to prove that, under a suitable extra condition, there is no closed eigenform for the Laplacian on such family.
{"title":"New Examples Of Shrinking Laplacian Solitons","authors":"Marina Nicolini","doi":"10.1093/qmath/haab029","DOIUrl":"https://doi.org/10.1093/qmath/haab029","url":null,"abstract":"We give a one-parameter family of examples of shrinking Laplacian solitons, which are the second known solutions to the closed $G_2$-Laplacian flow with a finite-time singularity. The torsion forms and the Laplacian and Ricci operators of a large family of $G_2$-structures on different Lie groups are also studied. We apply these formulas to prove that, under a suitable extra condition, there is no closed eigenform for the Laplacian on such family.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83684915","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 : 2020-06-01DOI: 10.1142/S0219887820501005
R. Kaushal, R. Sachdeva, Rakesh Kumar, R. K. Nagaich
We study generic Riemannian submersions from nearly Kaehler manifolds onto Riemannian manifolds. We investigate conditions for the integrability of various distributions arising for generic Riemannian submersions and also obtain conditions for leaves to be totally geodesic foliations. We obtain conditions for a generic Riemannian submersion to be a totally geodesic map and also study generic Riemannian submersions with totally umbilical fibers. Finally, we derive conditions for generic Riemannian submersions to be harmonic map.
{"title":"Semi-invariant Riemannian submersions from nearly Kaehler manifolds","authors":"R. Kaushal, R. Sachdeva, Rakesh Kumar, R. K. Nagaich","doi":"10.1142/S0219887820501005","DOIUrl":"https://doi.org/10.1142/S0219887820501005","url":null,"abstract":"We study generic Riemannian submersions from nearly Kaehler manifolds onto Riemannian manifolds. We investigate conditions for the integrability of various distributions arising for generic Riemannian submersions and also obtain conditions for leaves to be totally geodesic foliations. We obtain conditions for a generic Riemannian submersion to be a totally geodesic map and also study generic Riemannian submersions with totally umbilical fibers. Finally, we derive conditions for generic Riemannian submersions to be harmonic map.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72991564","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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