Department of Pure Mathematics (2007 - Present)
Pure mathematics - geometry
, Amirkabir University of Technology,
Pure Mathematics
, Amirkabir University of Technology,
Pure Mathematics
, Amirkabir University of Technology,
Associate professor, Department of Pure mathematics, Tarbiat Modares University. I got my B.Sc (1998), M.Sc (2000), and PhD(2006) in Pure Mathematics from Amirkabir University of Technology(Tehran Polytechnic). Since 2007 I am a faculty member of Pure Mathematics at the Tarbiat Modares University.
We study infinite-dimensional representations of Lie hypergroups on topological vector spaces and the corresponding smooth vectors. We find necessary and sufficient conditions for representations on the algebra of continuous functions with compact support to have continuous extensions to representations of the hypergroup. We prove an analog of the Bruhat theorem on smooth representations of Lie hypergroups.
In this article, we study isometric immersions of nearly K?hler manifolds into a space form (especially Euclidean space) and show that every nearly K?hler submanifold of a space form has an umbilic foliation whose leafs are 6-dimensional nearly K?hler manifolds. Moreover, using this foliation we show that there is no non-homogeneous 6-dimensional nearly K?hler submanifold of a space form. We prove some results towards a classification of nearly K?hler hypersurfaces in standard space forms.
In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distribution. The induced Poisson structure on the base manifold can be represented by means of the induced Poisson structures on the integral submanifolds. Moreover, for any compatible triple with invariant metric and admissible almost complex structure, we show that the bracket annihilates on the kernel of the anchor map.Subjects: Differential Geometry (math. DG)MSC classes: Primary 53C15, Secondary 53D17Cite as: arXiv: 1607.02907 [mat
Providing an appropriate definition of a horizontal subbundle of a Lie algebroid will lead to construction of a better framework on Lie algebriods. In this paper, we give a new and natural definition of a horizontal subbundle using the prolongation of a Lie algebroid and then we show that any linear connection on a Lie algebroid generates a horizontal subbundle and vice versa. The same correspondence will be proved for any covariant derivative on a Lie algebroid.
We study a class of Finsler metrics which contains the class of P-reducible metrics. Finsler metrics in this class are called generalized P-reducible metrics. We consider generalized P-reducible metrics with scalar flag curvature and find a condition under which these metrics reduce to C-reducible metrics. This generalizes Matsumoto’s theorem, which describes the equivalency of C-reducibility and P-reducibility on Finsler manifolds with scalar curvature. Then we show that generalized P-reducible metrics with vanishing stretch curvature are C-reducible.
We study the double cosets of a Lie group by a compact Lie subgroup. We show that a Weil formula holds for double coset Lie hypergroups and show that certain representations of the Lie group lift to representations of the double coset Lie hypergroup.
Statistics, as one of the applied sciences, has great impacts in vast area of other sciences. Prediction of protein structures with great emphasize on their geometrical features using dihedral angles has invoked the new branch of statistics, known as directional statistics. One of the available biological techniques to predict is molecular dynamics simulations producing high-dimensional molecular structure data. Hence, it is expected that the principal component analysis (PCA) can response some related statistical problems particulary to reduce dimensions of the involved variables. Since the dihedral angles are variables on non-Euclidean space (their locus is the torus), it is expected that direct implementation of PCA does not provide grea
We study the foliation space of complex and invariant (by torsion of intrinsic Hermitian connection) umbilic distribution on an isometric immersion from a nearly K?hler manifold into the Euclidean space. Under suitable conditions this leaf space is nearly K?hler and can be decomposed into a product of this leaf space and a 6-dimensional locally homogeneous nearly K?hler manifold.
We define Lie hypergroups and study their embedded and immersed subhypergroups. In particular we investigate the properties of the connected component of the identity, the universal covering and fundamental group of a Lie hypergroup. We also study the quotients and orbits in a Lie hypergroup.
In this paper, we study a class of Finsler metrics which contains the class of P-reducible metrics. Finsler metrics in this class are called generalized P-reducible metrics. We consider generalized P-reducible metrics with scalar flag curvature and find a condition under which these metrics reduce to C-reducible metrics. This generalize Matsumotos theorem, which describes the equivalency of C-reducibility and P-reducibility on Finsler manifolds with scalar curvature. Then we show that generalized P-reducible metrics with vanishing stretch curvature are C-reducible.Subjects: Differential Geometry (math. DG)MSC classes: 53C60, 53C25Cite as: arXiv: 1305.4629 [math. DG](or arXiv: 1305.4629 v1 [math. DG] for this version)Submission historyFrom:
The complete lift of a Riemannian metric g on a differentiable manifold M is not 0-homogeneous on the fibers of the tangent bundle TM. In this paper we introduce a new kind of lift G of g, which is 0-homogeneous. It determines a pseudo-Riemannian metric on , which depends only on the metric g. We obtain the Levi-Civita connection of this metric and study conformal vector fields on (). Finally, we introduce the almost product and complex structures which preserve homogeneity and study certain geometrical properties of these structures.
We introduce a class of metrics on the tangent bundle of a Riemannian manifold and find the Levi-Civita connections of these metrics. Then by using the Levi-Civita connection, we study the conformal vector fields on the tangent bundle of the Riemannian manifold. Finally, we obtain some relations between the flatness (resp. local symmetry) properties of the tangent bundle and the flatness (resp. local symmetry) on the base manifold.
Here, we present a new complete lift metric for which every infinitesimal fiber-preserving conformal transformation on the tangent bundle induces an infinitesimal projective transformation on the base manifold. Moreover, this correspondence gives rise to a homomorphism between Lie algebras. Also, we introduce an almost product structure on the tangent bundle and show that it is a product structure if and only if the corresponding Riemannian metric is of constant curvature.
The Lie derivation of multivector fields along multivector fields has been introduced by Schouten (see cite {Sc, S}), and studdied for example in cite {M} and cite {I}. In the present paper we define the Lie derivation of differential forms along multivector fields, and we extend this concept to covariant derivation on tangent bundles and vector bundles, and find natural relations between them and other familiar concepts. Also in spinor bundles, we introduce a covariant derivation along multivector fields and call it the Clifford covariant derivation of that spinor bundle, which is related to its structure and has a natural relation to its Dirac operator.
Recently, we have used the symmetric bracket of vector fields, and developed the notion of the symmetric derivation. Using this machinery, we have defined the concept of symmetric curvature. This concept is natural and is related to the notions divergence and Laplacian of vector fields. This concept is also related to the derivations on the algebra of symmetric forms which has been discussed by the authors. We introduce a new class of geometric vector fields and prove some basic facts about them. We call these vector fields affinewise. By contraction of the symmetric curvature, we define two new curvatures which have direct relations to the notions of divergence, Laplacian, and the Ricci tensor.
We introduce a pseudo-Riemannin metric on the cotangent bundle of a Riemannian manifold and show that the cotangent bundle with this metric is locally symmetric Einstein manifold. Also we obtain a locally symmetric Para-K?hler Einstein structure on the cotangent bundle of a Riemannian manifold of positive constant sectional curvature. Similar results are obtained on a tube around zero section in the cotangent bundle, in the case of a Riemannian manifold of negative constant sectional curvature.
Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometers using some Riemannian and pseudo-Riemannian lift metrics on TM. Here we consider the Riemannian or pseudo-Riemannian lift metric G on TM which is in some senses more general than other lift metrics previously defined on TM, and seems to complete these works. Next we study the lift conformal vector fields on (TM, G).
On a Finsler manifold, we define conformal vector fields and their complete lifts and prove that in certain conditions they are homothetic.
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