An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds.The present paper has two parts.The first part investigates topology of the isoparametric fa...An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds.The present paper has two parts.The first part investigates topology of the isoparametric families,namely the homotopy,homeomorphism,or diffeomorphism types,parallelizability,as well as the Lusternik-Schnirelmann category.This part extends substantially the results of Wang(J Differ Geom 27:55-66,1988).The second part is concerned with their curvatures;more precisely,we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.展开更多
in this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles o...in this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact non-negative curved manifolds admit （complete） metrics with non-negative curvature.展开更多
Let R be a right coherent ring and D^b（R-Mod） the bounded derived category of left R-modules. Denote by D^b（R-Mod）[GF,C] the subcategory of D^b（R-Mod） consisting of all complexes with both finite Gorenstein flat...Let R be a right coherent ring and D^b（R-Mod） the bounded derived category of left R-modules. Denote by D^b（R-Mod）[GF,C] the subcategory of D^b（R-Mod） consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K^b（F∩C） the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category D^b（R-Mod）[GF,C]/K^b（F∩C,） is triangle-equivalent to the stable category GF∩C of the Frobenius category of all Gorenstein fiat and cotorsion left R-modules.展开更多
基金partially supported by the NSFC(Nos.11722101,11871282,11931007)BNSF(Z190003)+1 种基金Nankai Zhide FoundationBeijing Institute of Technology Research Fund Program for Young Scholars.
文摘An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds.The present paper has two parts.The first part investigates topology of the isoparametric families,namely the homotopy,homeomorphism,or diffeomorphism types,parallelizability,as well as the Lusternik-Schnirelmann category.This part extends substantially the results of Wang(J Differ Geom 27:55-66,1988).The second part is concerned with their curvatures;more precisely,we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.
文摘in this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact non-negative curved manifolds admit （complete） metrics with non-negative curvature.
基金Supported by National Natural Science Foundation of China(Grant Nos.11601433 and 11261050)the Postdoctoral Science Foundation of China(Grant No.2106M602945XB)Northwest Normal University(Grant No.NWNU-LKQN-15-12)
文摘Let R be a right coherent ring and D^b（R-Mod） the bounded derived category of left R-modules. Denote by D^b（R-Mod）[GF,C] the subcategory of D^b（R-Mod） consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K^b（F∩C） the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category D^b（R-Mod）[GF,C]/K^b（F∩C,） is triangle-equivalent to the stable category GF∩C of the Frobenius category of all Gorenstein fiat and cotorsion left R-modules.
