Pub Date : 2020-10-01DOI: 10.4310/CAG.2020.V28.N8.A7
Brian Grajales, L. Grama, Caio J. C. Negreiros
We describe the invariant metrics on real flag manifolds and classify those with the following property: every geodesic is the orbit of a one-parameter subgroup. Such a metric is called g.o. (geodesic orbit). In contrast to the complex case, on real flag manifolds the isotropy representation can have equivalent submodules, which makes invariant metrics depend on more parameters and allows us to find more cases in which non-trivial g.o. metrics exist.
{"title":"Geodesic orbit spaces in real flag manifolds","authors":"Brian Grajales, L. Grama, Caio J. C. Negreiros","doi":"10.4310/CAG.2020.V28.N8.A7","DOIUrl":"https://doi.org/10.4310/CAG.2020.V28.N8.A7","url":null,"abstract":"We describe the invariant metrics on real flag manifolds and classify those with the following property: every geodesic is the orbit of a one-parameter subgroup. Such a metric is called g.o. (geodesic orbit). In contrast to the complex case, on real flag manifolds the isotropy representation can have equivalent submodules, which makes invariant metrics depend on more parameters and allows us to find more cases in which non-trivial g.o. metrics exist.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85762173","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}
A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line.
{"title":"Closed Affine Manifolds with an Invariant Line","authors":"Charles Daly","doi":"10.13016/RRHO-KQRY","DOIUrl":"https://doi.org/10.13016/RRHO-KQRY","url":null,"abstract":"A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"467 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79888509","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}
In this survey, symmetry provides a framework for classification of manifolds with differential-geometric structures. We highlight pseudo-Riemannian metrics, conformal structures, and projective structures. A range of techniques have been developed and successfully deployed in this subject, some of them based on algebra and dynamics and some based on analysis. We aim to illustrate this variety.
{"title":"Rigidity of Transformation Groups in Differential Geometry","authors":"K. Melnick","doi":"10.1090/NOTI2279","DOIUrl":"https://doi.org/10.1090/NOTI2279","url":null,"abstract":"In this survey, symmetry provides a framework for classification of manifolds with differential-geometric structures. We highlight pseudo-Riemannian metrics, conformal structures, and projective structures. A range of techniques have been developed and successfully deployed in this subject, some of them based on algebra and dynamics and some based on analysis. We aim to illustrate this variety.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82069547","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 : 2020-09-28DOI: 10.1142/S0129167X21500221
Shunya Fujii, S. Maeta
In this paper, we consider generalized Yamabe solitons which include many notions, such as Yamabe solitons, almost Yamabe solitons, h-almost Yamabe solitons, gradient k-Yamabe solitons and conformal gradient solitons. We completely classify the generalized Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field.
{"title":"Classification of generalized Yamabe solitons in Euclidean spaces","authors":"Shunya Fujii, S. Maeta","doi":"10.1142/S0129167X21500221","DOIUrl":"https://doi.org/10.1142/S0129167X21500221","url":null,"abstract":"In this paper, we consider generalized Yamabe solitons which include many notions, such as Yamabe solitons, almost Yamabe solitons, h-almost Yamabe solitons, gradient k-Yamabe solitons and conformal gradient solitons. We completely classify the generalized Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"111 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73353854","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}
In this short note, we prove a uniqueness result for small entropy self-expanders asymptotic to a fixed cone. This is a direct consequence of the mountain-pass theorem and the integer degree argument proved by J. Bernstein and L. Wang.
{"title":"A uniqueness result for self-expanders with small entropy","authors":"Junfu Yao","doi":"10.1090/proc/15862","DOIUrl":"https://doi.org/10.1090/proc/15862","url":null,"abstract":"In this short note, we prove a uniqueness result for small entropy self-expanders asymptotic to a fixed cone. This is a direct consequence of the mountain-pass theorem and the integer degree argument proved by J. Bernstein and L. Wang.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81812933","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 extend results of Llarull and Goette-Semmelmann to manifolds with boundary.
我们将Llarull和gotte - semmelmann的结果推广到有边界的流形。
{"title":"Index theory for scalar curvature on manifolds with boundary","authors":"J. Lott","doi":"10.1090/PROC/15551","DOIUrl":"https://doi.org/10.1090/PROC/15551","url":null,"abstract":"We extend results of Llarull and Goette-Semmelmann to manifolds with boundary.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"150 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77854632","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 : 2020-09-15DOI: 10.2991/JNMP.K.210419.001
Sining Wei, Yong Wang
In this paper, we give the Lichnerowicz type formulas for statistical de Rham Hodge operators. Moreover, Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with (respectively without) boundary are proved.
{"title":"Statistical de Rham Hodge Operators and the Kastler-Kalau-Walze Type Theorem for Manifolds With Boundary","authors":"Sining Wei, Yong Wang","doi":"10.2991/JNMP.K.210419.001","DOIUrl":"https://doi.org/10.2991/JNMP.K.210419.001","url":null,"abstract":"In this paper, we give the Lichnerowicz type formulas for statistical de Rham Hodge operators. Moreover, Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with (respectively without) boundary are proved.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"61 15","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91514084","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 : 2020-09-13DOI: 10.17398/2605-5686.36.1.99
S. Anastassiou, I. Chrysikos
For any flag manifold $M=G/K$ of a compact simple Lie group $G$ we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions pass through an invariant Einstein metric on $M$, and by a result of Bohm-Lafuente-Simon ([BoLS17]) they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold $M=G/K$ with second Betti number $b_{2}(M)=1$, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose $alpha$-limit set consists of fixed points at infinity of $mathscr{M}^G$. We show that these fixed points correspond to invariant Einstein metrics and based on the Poincare compactification method, we study their stability properties, illuminating thus the structure of the system's phase space.
{"title":"Ancient solutions of the homogeneous Ricci flow on flag manifolds","authors":"S. Anastassiou, I. Chrysikos","doi":"10.17398/2605-5686.36.1.99","DOIUrl":"https://doi.org/10.17398/2605-5686.36.1.99","url":null,"abstract":"For any flag manifold $M=G/K$ of a compact simple Lie group $G$ we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions pass through an invariant Einstein metric on $M$, and by a result of Bohm-Lafuente-Simon ([BoLS17]) they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold $M=G/K$ with second Betti number $b_{2}(M)=1$, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose $alpha$-limit set consists of fixed points at infinity of $mathscr{M}^G$. We show that these fixed points correspond to invariant Einstein metrics and based on the Poincare compactification method, we study their stability properties, illuminating thus the structure of the system's phase space.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83477384","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}
In this note we show that a closed Riemannian manifold does not admit a fill-in with nonnegative scalar curvature if the mean curvature is point-wise large. Similar result also holds for fill-ins with a negative scalar curvature lower bound.
{"title":"Nonexistence of NNSC fill-ins with large mean curvature","authors":"P. Miao","doi":"10.1090/proc/15400","DOIUrl":"https://doi.org/10.1090/proc/15400","url":null,"abstract":"In this note we show that a closed Riemannian manifold does not admit a fill-in with nonnegative scalar curvature if the mean curvature is point-wise large. Similar result also holds for fill-ins with a negative scalar curvature lower bound.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"178 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77367935","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 : 2020-09-01DOI: 10.1142/S0219199721500358
Marco Ghimenti, A. Micheletti
We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n>6 and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.
{"title":"Blowing up solutions for supercritical Yamabe problems on manifolds with non-umbilic boundary","authors":"Marco Ghimenti, A. Micheletti","doi":"10.1142/S0219199721500358","DOIUrl":"https://doi.org/10.1142/S0219199721500358","url":null,"abstract":"We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n>6 and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73137656","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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