In this paper,we study locally strongly convex affine hyperspheres in the unimodular affine space Rn+1 which,as Riemannian manifolds,are locally isometric to the Riemannian product of two Riemannian manifolds both pos...In this paper,we study locally strongly convex affine hyperspheres in the unimodular affine space Rn+1 which,as Riemannian manifolds,are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature.As the main result,a complete classification of such affine hyperspheres is established.Moreover,as direct consequences,3-and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified.展开更多
In this paper, we study Lagrangian submanifolds of the nearly Kaehler 6-sphere. We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic subman...In this paper, we study Lagrangian submanifolds of the nearly Kaehler 6-sphere. We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic submanifold. Our pinching result improves a previous result obtained by H. Li.展开更多
We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angl...We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angle functions encoding the geometry of the Lagrangian submanifold at hand.We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface.We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions,respectively all but one,coincide.展开更多
We investigate n-dimensional(n≥4),conformally flat,minimal,Lagrangian submanifolds of the n-dimensional complex space form in terms of the multiplicities of the eigenvalues of the Schouten tensor and classify those t...We investigate n-dimensional(n≥4),conformally flat,minimal,Lagrangian submanifolds of the n-dimensional complex space form in terms of the multiplicities of the eigenvalues of the Schouten tensor and classify those that admit at most one eigenvalue of multiplicity one.In the case where the ambient space is Cn,the quasi umbilical case was studied in Blair(2007).However,the classification there is not complete and several examples are missing.Here,we complete(and extend) the classification and we also deal with the case where the ambient complex space form has non-vanishing holomorphic sectional curvature.展开更多
In this paper, continuing with Hu-Li Vrancken and the recent work of Antid Dillen- Schoels-Vrancken, we obtain a decomposition theorem which settled the problem of how to determine whether a given locally strongly con...In this paper, continuing with Hu-Li Vrancken and the recent work of Antid Dillen- Schoels-Vrancken, we obtain a decomposition theorem which settled the problem of how to determine whether a given locally strongly convex aitine hypersurface can be decomposed as a generalized Calabi composition of two affine hyperspheres, based on the properties of its difference tensor K and its affine shape operator S.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11771404)。
文摘In this paper,we study locally strongly convex affine hyperspheres in the unimodular affine space Rn+1 which,as Riemannian manifolds,are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature.As the main result,a complete classification of such affine hyperspheres is established.Moreover,as direct consequences,3-and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified.
基金Research partially supported by the Ministry of Science and Environmental Protectipn of Serbia, Project 1646Research partially supported by EGIDE, Pavle Savic 07945VC(France)
文摘In this paper, we study Lagrangian submanifolds of the nearly Kaehler 6-sphere. We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic submanifold. Our pinching result improves a previous result obtained by H. Li.
基金supported by the Tsinghua University-KU Leuven Bilateral Scientific Cooperation Fundcollaboration project funded by National Natural Science Foundation of China+6 种基金supported by National Natural Science Foundation of China(Grant Nos.11831005 and 11671224)supported byNational Natural Science Foundation of China(Grant Nos.11831005 and 11671223)supported by National Natural Science Foundation of China(Grant No.11571185)the Research Foundation Flanders(Grant No.11961131001)supported by the Excellence of Science Project of the Belgian Government(Grant No.GOH4518N)supported by the KU Leuven Research Fund(Grant No.3E160361)the Fundamental Research Funds for the Central Universities。
文摘We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angle functions encoding the geometry of the Lagrangian submanifold at hand.We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface.We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions,respectively all but one,coincide.
基金supported by the Ministry of Education,Science and Technological Development of the Republic of Serbia(Grant No.174012)。
文摘We investigate n-dimensional(n≥4),conformally flat,minimal,Lagrangian submanifolds of the n-dimensional complex space form in terms of the multiplicities of the eigenvalues of the Schouten tensor and classify those that admit at most one eigenvalue of multiplicity one.In the case where the ambient space is Cn,the quasi umbilical case was studied in Blair(2007).However,the classification there is not complete and several examples are missing.Here,we complete(and extend) the classification and we also deal with the case where the ambient complex space form has non-vanishing holomorphic sectional curvature.
基金supported by the Ministry of Science and Technological Development of Serbia,Pro ject174012supported by NSFC(Grant No.11371330)supported by NSFC(Grant Nos.11326072 and 11401173)
文摘In this paper, continuing with Hu-Li Vrancken and the recent work of Antid Dillen- Schoels-Vrancken, we obtain a decomposition theorem which settled the problem of how to determine whether a given locally strongly convex aitine hypersurface can be decomposed as a generalized Calabi composition of two affine hyperspheres, based on the properties of its difference tensor K and its affine shape operator S.