Sharief Deshmukh

Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature

AbstractLet (M, g) be an n-dimensional compact and connected Riemannian manifold of constant scalar curvature. If the sectional curvatures of M are bounded below by a constant α > 0, and the Ricci curvature satisfies Ric < (n − 1)αδ, δ ≥ 1, then it is shown that either M is isometric to the n-sphere Sn(α) or else each nonzero eigenvalue λ of the Laplacian acting on the smooth functions of M satisfies the following: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae83UdW% 2aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG4maiaad6gacqaHXoqy% caGGOaGae8hTdqMaeyOeI0IaaGOmaiaacMcacqWF7oaBcqWFRaWkcq% WFYaGmcaWGUbGaeqySde2aaWbaaSqabeaacaaIYaaaaOGae8hTdqMa% e8hkaGIae8xmaeJae83kaSIae8hkaGccbiGaa4NBaiab-jHiTiab-f% daXiab-LcaPiab-r7aKjab-LcaPiab-5da+iab-bdaWaaa!556A!

Real hypersurfaces of ⁿ with non-negative Ricci curvature

\lambda ^2 + 3n\alpha (\delta - 2)\lambda + 2n\alpha ^2 \delta (1 + (n - 1)\delta ) > 0

CLASSIFICATION OF RICCI SOLITONS ON EUCLIDEAN HYPERSURFACES

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A note on trans-Sasakian manifolds

The coupling of electromagnetic fields for the construction of light amplifiers is described by a certain Hamiltonian which can be expressed as the sum of three differential operators. Following the method of Steinberg (1977) and Dattoli et. al. (1990), the solution of Schrodingers equation for the Hamiltonian of the system is constructed from the solutions of the corresponding equations for the components of the Hamiltonian.

Yamabe and Quasi-Yamabe Solitons on Euclidean Submanifolds

We prove the non-existence of Levi flat compact real hypersurfaces without boundary in CPn,n > 1, with non-negative totally real Ricci curvature.

Real hypersurfaces of a complex space form

All above theorems are consequences of an integral formula which we prove in Section 2. We observe that Theorem 4 generalizes Theorem 1 in [1] for hypersurfaces of non-negative Ricci curvature in a Euclidean space. We also get the following corollary to Theorem 1 which generalizes the result of Jacobwitz [2] for non-immersibility of a compact Riemannian manifold of non-negative Ricci curvature into a closed ball in a Euclidean space.

Compact manifolds of constant scalar curvature

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

Almost complex surfaces in the nearly Kaehler S6

In this paper, we obtain some sufficient conditions for a 3-dimensional compact trans-Sasakian manifold of type (α, β) to be homothetic to a Sasakian manifold. A characterization of a 3-dimensional cosymplectic manifold is also obtained.