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Luc Vrancken - One of the best experts on this subject based on the ideXlab platform.
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minimal lagrangian submanifolds with Constant Sectional Curvature in indefinite complex space forms
Proceedings of the American Mathematical Society, 2002Co-Authors: Luc VranckenAbstract:We study minimal Lagrangian immersions from an indefinite real space form M n s (c) into an indefinite complex space form M n s (4c). Provided that c ¬= c, we show that M n s (c) has to be flat and we obtain an explicit description of the immersion. In the case when the metric is positive definite or Lorentzian, this result was respectively obtained by Ejiri (1982) and by Kriele and the author (1999). In the case that c = c, this theorem is no longer true; see for instance the examples discovered by Chen and the author (accepted for publication in the Tohoku Mathematical Journal).
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lorentzian affine hyperspheres with Constant affine Sectional Curvature
Transactions of the American Mathematical Society, 2000Co-Authors: Marcus Kriele, Luc VranckenAbstract:We study ane hyperspheres M with Constant Sectional Curvature (with respect to the ane metric h). A conjecture by M. Magid and P. Ryan states that every such ane hypersphere with nonzero Pick invariant is anely equivalent to either where the dimension n satises n =2 m 1o rn =2 m .U p to now, this conjecture was proved if M is positive denite or if M is a 3-dimensional Lorentz space. In this paper, we give an armative answer to this conjecture for arbitrary dimensional Lorentzian ane hyperspheres.
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minimal lagrangian submanifolds of lorentzian complex space forms with Constant Sectional Curvature
Archiv der Mathematik, 1999Co-Authors: Marcus Kriele, Luc VranckenAbstract:We study Lagrangian immersions \(M^n_1\) into Lorentzian complex indefinite space forms \(\tilde M^n_1(4 \tilde c), \tilde c \ne 0\) and classify all such immersions which are minimal and have Constant Curvature \(c \ne \tilde c\).
Opozda Barbara - One of the best experts on this subject based on the ideXlab platform.
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Curvature bounded conjugate symmetric statistical structures with complete metric
'Springer Science and Business Media LLC', 2019Co-Authors: Opozda BarbaraAbstract:In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about Constant Sectional Curvature is replaced by the assumption that the Curvature satisfies some inequalities
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Curvature bounded conjugate symmetric statistical structures with complete metric
2018Co-Authors: Opozda BarbaraAbstract:In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about Constant Sectional Curvature is replaced by the assumption that the Curvature satisfies some inequalities.Comment: 15 page
D. D. Porosniuc - One of the best experts on this subject based on the ideXlab platform.
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a locally symmetric kaehler einstein structure on a tube in the nonzero cotangent bundle of a space form
arXiv: Differential Geometry, 2003Co-Authors: D. D. PorosniucAbstract:We obtain a locally symmetric Kaehler Einstein structure on a tube in the nonzero cotangent bundle of a Riemannian manifold of positive Constant Sectional Curvature. The obtained Kaehler Einstein structure cannot have Constant holomorphic Sectional Curvature.
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a kaehler einstein structure on the cotangent bundle of a riemannian manifold
arXiv: Differential Geometry, 2003Co-Authors: Vasile Oproiu, D. D. PorosniucAbstract:We use the natural lifts of the fundamental tensor field g to the cotangent bundle T*M of a Riemannian manifold (M,g), in order to construct an almost Hermitian structure (G,J) of diagonal type on T*M. The obtained almost complex structure J on T*M is integrable if and only if the base manifold has Constant Sectional Curvature and the second coefficient, involved in its definition is expressed as a rational function of the first coefficient and its first order derivative. Next one shows that the obtained almost Hermitian structure is almost Kaehlerian. Combining the obtained results we get a family of Kaehlerian structures on T*M, depending on one essential parameter. Next we study the conditions under which the considered Kaehlerian structure is Einstein. In this case (T*M,G,J) has Constant holomorphic Curvature.
Vrancken Luc - One of the best experts on this subject based on the ideXlab platform.
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On product minimal Lagrangian submanifolds in complex space forms
'Springer Science and Business Media LLC', 2019Co-Authors: Cheng Xiuxiu, Hu Zejun, Moruz Marilena, Vrancken LucAbstract:In this paper we consider minimal Lagrangian submanifolds in $n$-dimensional complex space forms. More precisely, we study such submanifolds which, endowed with the induced metrics, write as a Riemannian product of two Riemannian manifolds, each having Constant Sectional Curvature. As the main result, we give a complete classification of these submanifolds
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Lagrangian Submanifolds with Constant Angle functions of the Nearly Kähler S3 × S3
'Elsevier BV', 2018Co-Authors: Bektaş Burcu, Moruz Marilena, Veken, Joeri Van Der, Vrancken LucAbstract:We study Lagrangian submanifolds of the nearly Kähler S3 × S3 with respect to their so called angle functions. We show that if all angle functions are Constant, then the submanifold is either totally geodesic or has Constant Sectional Curvature and there is a classification theorem that follows from Dioos et al. (2018). Moreover, we show that if precisely one angle function is Constant, then it must be equal to 0, π 3 or 2π 3 . Using then two remarkable constructions together with the classification of Lagrangian submanifolds of which the first component has nowhere maximal rank from, Bektaş et al. (2018), we obtain a classification of such Lagrangian submanifolds
Wayne Rossman - One of the best experts on this subject based on the ideXlab platform.
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a loop group formulation for Constant Curvature submanifolds of pseudo euclidean space
Taiwanese Journal of Mathematics, 2008Co-Authors: David Brander, Wayne RossmanAbstract:We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudo-Riemannian manifold $M_{c,r}^m$, of dimension $m$, Constant Sectional Curvature $c \neq 0$, and signature $r$, into the pseudo-Euclidean space $\bf R_s^{m+k}$, of signature $s\geq r$. In fact these immersions are obtained canonically from the loop group maps corresponding to isometric immersions of the same manifold into a pseudo-Riemannian sphere or hyperbolic space $S_s^{m+k}$ or $H_s^{m+k}$, which have been known for some time. A simple formula is given for obtaining these immersions from those loop group maps.
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a loop group formulation for Constant Curvature submanifolds of pseudo euclidean space
arXiv: Differential Geometry, 2006Co-Authors: David Brander, Wayne RossmanAbstract:We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudo-Riemannian manifold $M_{c,r}^m$, of dimension $m$, Constant Sectional Curvature $c \neq 0$, and signature $r$, into the pseudo-Euclidean space $\real_s^{m+k}$, of signature $s\geq r$. In fact these immersions are obtained canonically from the loop group maps corresponding to isometric immersions of the same manifold into a pseudo-Riemannian sphere or hyperbolic space $S_s^{m+k}$ or $H_s^{m+k}$, which have previously been studied. A simple formula is given for obtaining these immersions from those loop group maps.