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تحميل الدليل التدريبي

أسئلة شائعة


 

 

Main Areas of Research Interest: I am interested in some specific aspects of the following areas

(1) Differential Geometry, (2) Geometric Analysis, (3) Representations of Lie groups and Differential equations, (4) Einstein's Field equation.

Specifics of Research Interest:

(1) Differential Geometry: Differential Geometry is a very vast field and contains hundredes of sub fields. My research interests are in the following sub fields with some humble contributions:

(i) Geometry of submanifolds: In geometry of submanifolds of a Riemannian manifolds my work consists of following special areas:

(a) Minimal submanifolds of a unit sphere. [1], [2], [3], [4], [5] 

(b) Submanifolds with restriction on  curvature. [1], [2], [3]

(c) Hypersurfaces of real space form. [1], [2] 

(d) Real hypersurfaces of complex space form. [1], [2], [3], [4] [5]

(e) CR-submanifolds of a Almost Hermitian manifolds. [1], [2], [3], [4], [5]

(f) Characterization of Spheres. [1], [2], [3], [4]

(ii) Non-immersibility in Euclidean spaces: After the work of Nash that a compact Riemannian manifold can be isometrically immersed in a Euclidean space of suffieciently high dimension, the question of non-immersibility was taken up. In this direction the first result is of Chern who proved that non-positively curved  n-dimensional Riemannian manifold can not be isometrically immersed in a Euclidean space of dimension (2n-1) and this result was generalized by Jacowicz who proved that an n-dimensional compact Riemannian manifold whose sectional curvatures are strictly less than a positive number k then no isometric immersion of it in a Euclidean space of dimension (2n-1) is contained a ball of radius 1/k. All these non-immersibility resuts involve bounds on sectional curvature, we are interested in obtaining bounds on Ricci curvature or scalar curvature of the Riemannian manifold in question. [1], [2], [3]

(iii) Riemannian submersions: Riemannian submersions is a dual concept to the isometric immersion and naturally gives rise to two distributions on the Riemannian manifold submersed on to a base Riemannian manifold. Kobayashi realised that a CR-Submanifold of a Kaehler manifold also has two complementary smooth distributions and initiated the study of submersions of CR-Submanifold onto almost Hermitian manifolds. We extended the results of Kobayashi on submersions of CR-Submanifolds of a Kaehler manifold as well those of other classes of Almost Hermitian manifolds. [1], [2]

(iv) Geometry of Tangent bundle: Geometry of Tangent bundle of a Riemannian manifold with Sasaki metric has been quite extensively studied since last four decades. However, the natural complex structure on the tangent bundle, though compatible with the Sasakian metric, does not have nice complex geometry unless the base manifold is flat. This necessiated a need for search of other Riemannian metrics on the Tangent bundle and there is a good amount of study in this direction. We are interested in the tangent bundle of the hypersurface of a Eulcidean space which itself becomes submanifold of a Eulcidean space and therefore has naturally induced Riemannian metric and were able to get some intersting results. [1], [2] 

(v) Conformal Geometry: In conformal geometry, we are interested in the folllowing special areas:

(a) Finding necessary and sufficient condition for a conformal transformation on a Riemannian manifold to be a homothety.

(b) Effect of the existence of conformal vector fields on the geometry of a Riemannian manifold. [1], [2] 

(vi) Structures on a Riemannian manifold: Apart from Almost Hermitian Geometry, we had interest in contact structures, Sasakian structures, Paracontact structures on a Riemannian manifold and their influence on the geometry of the Riemannian manifold. [1], [2], [3]

(2) Geometric Analysis: Geometric Analysis is again a very vast area and quite demanding, with our limited capacity we work on the small special aspects of the following major areas:

(i) Spectrum of Laplace operator: We are interested in estimating the bounds on the eigenvalues of the Laplacian operator acting on smooth functions on a Riemannian manifold using the curvature information of the Riemannian manifold. [1], [2], [3] [4], [5]

(ii) Ricci Solitons: Ricci solitons are self siimlar solutions of the Ricci flow on a Riemannian manifold and are very siginfican as they were employed by Pelerman to settle the century old Poincare Conjecture. After this famous conjecture is settled, the Ricci soliton have become subject of interest because of their applicability in physical sciences, medical sciences and engineering. Hamilton (the founder of Ricci flow) conjectured that a gradient Ricci soliton of positive curvature operator must be an Einstein manifold. Our interest in Ricci soliton lies in at least finding conditions under which this conjecture is true. [1], [2]

(3) Lie groups and differential equations: Given a differential equation it gives rise to linearly independent first order differential operators which satisfy commutation relations therefore generate a finite dimensional Lie-algebra which is isomorphic to Lie-algebra of a Lie-group. The representation of these Lie-groups in the space of analytic functions defined on the domain of the differential equation makes it possible to study the properties of the solutions of the differential equation. We had interest in the representation of the Lie-gourps SL(2), O(3) and study the properties of special functions as well as some PDE's. [1]

(4) Einstein's Field equation: Given a symmetric tensor T of type (0,2) on a smooth compact manifold M, we are interested studying conditions under which there exist a  Riemannian metric g on M that satisfies the Einstein's field equation

Ric-1/2Sg=T

where Ric is the Ricci curvature and S is the scalar curvature with respect to the metric g.[1], [2]

 

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