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Indranil Biswas - One of the best experts on this subject based on the ideXlab platform.
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on stability of Tangent Bundle of toric varieties
Proceedings - Mathematical Sciences, 2021Co-Authors: Indranil Biswas, Arijit Dey, Ozhan Genc, Mainak PoddarAbstract:Let $X$ be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the Tangent Bundle $T{X}$. In particular, a complete answer is given when $X$ is a Fano toric variety of dimension four with Picard number at most two, complementing earlier work of Nakagawa. We also give an infinite set of examples of Fano toric varieties for which $TX$ is unstable; the dimensions of this collection of varieties are unbounded. Our method is based on the equivariant approach initiated by Klyachko and developed further by Perling and Kool.
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homogeneous principal Bundles over manifolds with trivial logarithmic Tangent Bundle
arXiv: Complex Variables, 2019Co-Authors: H Azad, Indranil Biswas, Azeem M KhadamAbstract:Winkelmann considered compact complex manifolds $X$ equipped with a reduced effective normal crossing divisor $D\, \subset\, X$ such that the logarithmic Tangent Bundle $TX(-\log D)$ is holomorphically trivial. He characterized them as pairs $(X,\, D)$ admitting a holomorphic action of a complex Lie group $\mathbb G$ satisfying certain conditions \cite{Wi1}, \cite{Wi2}; this $\mathbb G$ is the connected component, containing the identity element, of the group of holomorphic automorphisms of $X$ that preserve $D$. We characterize the homogeneous holomorphic principal $H$--Bundles over $X$, where $H$ is a connected complex Lie group. Our characterization says that the following three are equivalent: (1)~ $E_H$ is homogeneous. (2)~ $E_H$ admits a logarithmic connection singular over $D$. (3)~ The family of principal $H$--Bundles $\{g^*E_H\}_{g\in \mathbb G}$ is infinitesimally rigid at the identity element of the group $\mathbb G$.
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Stability of the Tangent Bundle of G/P in positive characteristics
Algebras and Representation Theory, 2018Co-Authors: Indranil Biswas, Pierre-emmanuel Chaput, Christophe MourouganeAbstract:Let $G$ be an almost simple simply-connected affine algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. If $G$ has type $B_n$, $C_n$ or $F_4$, we assume that $p > 2$, and if $G$ has type $G_2$, we assume that $p > 3$. Let $P \subset G$ be a parabolic subgroup. We prove that the Tangent Bundle of $G/P$ is Frobenius stable with respect to any polarization on $G/P$.
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stability of the Tangent Bundle of g p in positive characteristics
arXiv: Algebraic Geometry, 2015Co-Authors: Indranil Biswas, Pierre-emmanuel Chaput, Christophe MourouganeAbstract:Let $G$ be an almost simple simply-connected affine algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. If $G$ has type $B_n$, $C_n$ or $F_4$, we assume that $p > 2$, and if $G$ has type $G_2$, we assume that $p > 3$. Let $P \subset G$ be a parabolic subgroup. We prove that the Tangent Bundle of $G/P$ is Frobenius stable with respect to the anticanonical polarization on $G/P$.
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essentially finite vector Bundles on varieties with trivial Tangent Bundle
Proceedings of the American Mathematical Society, 2011Co-Authors: Indranil Biswas, A J Parameswaran, S SubramanianAbstract:Let X be a smooth projective variety, defined over an algebraically closed field of positive characteristic, such that the Tangent Bundle TX is trivial. Let F X : X → X be the absolute Frobenius morphism of X. We prove that for any n ≥ 1, the n―fold composition F n X is a torsor over X for a finite group—scheme that depends on n. For any vector Bundle E → X, we show that the direct image (F n X ) * E is essentially finite (respectively, F—trivial) if and only if E is essentially finite (respectively, F―trivial).
Aydin Gezer - One of the best experts on this subject based on the ideXlab platform.
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Some harmonic problems on the Tangent Bundle with a Berger-type deformed Sasaki metric
arXiv: Differential Geometry, 2020Co-Authors: Murat Altunbas, Ramazan Simsek, Aydin GezerAbstract:Let $(M_{2k},\varphi ,g)$ be an almost anti-paraHermitian manifold and $(TM,g_{BS})$ be its Tangent Bundle with a Berger type deformed Sasaki metric $g_{BS}$. In this paper, we deal with the harmonicity of the canonical projection $\pi :TM\rightarrow M$ and a vector field $\xi $ which is considered as a map $\xi :M\rightarrow TM$.
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on the Tangent Bundle with deformed sasaki metric
International Electronic Journal of Geometry, 2013Co-Authors: Aydin GezerAbstract:In the present paper, we consider a deformation (in the horizontal Bundle) of Sasaki metric on the Tangent Bundle TM over an n dimensional Riemannian manifold (M;g): We rstly study some properties of deformed Sasaki metric which is pure with respect to some paracomplex structures on TM: Finally conditions for deformed Sasaki metric to be recurrent or pseudo symmetric are given.
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infinitesimal affine transformations in the Tangent Bundle of a riemannian manifold with respect to the horizontal lift of an affine connection
Hacettepe Journal of Mathematics and Statistics, 2006Co-Authors: Aydin Gezer, Kursat AkbulutAbstract:Infinitesimal Affine Transformations in the Tangent Bundle of a Riemannian Manifold with respect to the Horizontal Lift of an Affine Connection
Joerg Winkelmann - One of the best experts on this subject based on the ideXlab platform.
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on manifolds with trivial logarithmic Tangent Bundle the non kahler case
Transformation Groups, 2008Co-Authors: Joerg WinkelmannAbstract:We study non-Kahler manifolds with trivial logarithmic Tangent Bundle. We show that each such manifold arises as a fibre Bundle with a compact complex parallelizable manifold as basis and a compactficiation of a semi-torus as fibre.
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on manifolds with trivial logarithmic Tangent Bundle the non kaehler case
arXiv: Complex Variables, 2005Co-Authors: Joerg WinkelmannAbstract:We study non-Kaehler manifolds with trivial logarithmic Tangent Bundle. We show that each such manifold arises as a fiber Bundle with a compact complex parallelizable manifold as basis and a toric variety as fiber.
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on varieties with trivial logarithmic Tangent Bundle
arXiv: Algebraic Geometry, 2002Co-Authors: Joerg WinkelmannAbstract:We characterize Kaehler manifolds with trivial logarithmic Tangent Bundle (with respect to a divisor D) as a class of certain compatifications of complex semi-tori.
G E Prince - One of the best experts on this subject based on the ideXlab platform.
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Tangent Bundle geometry induced by second order partial differential equations
Journal de Mathématiques Pures et Appliquées, 2016Co-Authors: D J Saunders, Olga Rossi, G E PrinceAbstract:Abstract We show how the Tangent Bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs ‘of connection type’. Whereas for ODEs the decomposition is intrinsic, for PDEs it is necessary to specify a closed 1-form on the manifold of independent variables, together with a transverse local vector field. The resulting decomposition provides several natural curvature operators. We give three examples to indicate possible applications of this theory.
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Tangent Bundle geometry induced by second order partial differential equations
arXiv: Differential Geometry, 2014Co-Authors: D J Saunders, Olga Rossi, G E PrinceAbstract:We show how the Tangent Bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs `of connection type'. Whereas for ODEs the decomposition is intrinsic, for PDEs it is necessary to specify a closed 1-form on the manifold of independent variables, together with a transverse local vector field. The resulting decomposition provides several natural curvature operators. The harmonic map equation is examined, and in this case both the 1-form and the vector field arise naturally.
Fouzi Hathout - One of the best experts on this subject based on the ideXlab platform.
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N-Legendre and N-slant curves in the unit Tangent Bundle of Minkowski surfaces
Asian-european Journal of Mathematics, 2017Co-Authors: Murat Bekar, Fouzi HathoutAbstract:Let (T1𝕄12,g 1) be a unit Tangent Bundle of Minkowski surface (𝕄12,g) endowed with the pseudo-Riemannian induced Sasaki metric. In this present paper, we studied the N-Legendre and N-slant curves in which the inner product of its normal vector and Reeb vector is zero and nonzero constant, respectively, in T1𝕄12,g 1 and several important characterizations of these curves are given.
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ruled surfaces and Tangent Bundle of unit 2 sphere
International Journal of Geometric Methods in Modern Physics, 2017Co-Authors: Fouzi Hathout, Murat Bekar, Yusuf YayliAbstract:In this paper, a one-to-one correspondence is given between the Tangent Bundle of unit 2-sphere, T𝕊2, and the unit dual sphere, 𝕊𝔻2. According to Study’s map, to each curve on 𝕊𝔻2 corresponds a rul...
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N-Legendre and N-slant curves in the unit Tangent Bundle of surfaces
kuwait journal of science, 2017Co-Authors: Fouzi Hathout, Murat BekarAbstract:Let (T1M; g1) be a unit Tangent Bundle of some surface (M; g) en-dowed with the induced Sasaki metric. In this present paper, we de- ne two kinds of curves called N-legendre and N-slant curves as curveshaving an inner product of normal vector and Reeb vector zero andnonzero constant respectively and several important characterizationsof these curves are obtained.
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N-Legendre and N-Slant Curves in the Unit Tangent Bundle of Minkowski Surfaces
arXiv: Differential Geometry, 2016Co-Authors: Murat Bekar, Fouzi HathoutAbstract:Let $(\mathbb{M}_{1}^{2},g)$ be a Minkowski surface and $(T_1\mathbb{M}_1^2, g_1)$ its unit Tangent Bundle endowed with the pseudo-Riemannian induced Sasaki metric. We extend in this paper the study of the N-Legendre and N-slant curves which the inner product of normal vector and Reeb vector is zero and nonzero constant respectively in $\left( T_1 \mathbb{M}_1^2, g_1 \right)$, given in \cite{hmy}, to the Minkowski context and several important characterizations of these curves are given.\newline
