In this paper we prove two sharp inequalities involving the normalized scalar curvature and the generalized normalized δ-Casorati curvatures for slant submanifolds in quaternionic space forms. We also characterize those submanifolds for which the equality cases hold. These results are a generalization of some recent results concerning the Casorati curvature for a slant submanifold in a quaterni...

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvatu...

where ω̃ = ( √ −1/2)g̃ij̄dz ∧ dz and Σ̃ = ( √ −1/2)R̃ij̄dz ∧ dz are the Kähler form, the Ricci form of the metric g̃ respectively, while c1(M) denotes the first Chern class. Under the normalized initial condition (2), the first author [3] (see also Proposition 1.1 in [4]) showed that the solution g(x, t) = ∑ gij̄(x, t)dz dz to the normalized flow (1) exists for all time. Furthermore by the work of Mok ...

Let f : X −→ S be a smooth projective family and let (L, h) be a singular hermitian line bundle on X with semipositive curvature current. Let Ks := K(Xs, KXs +L | Xs, h | Xs)(s ∈ S) be the Bergman kernel of KXs +L | Xs with respect to h | Xs and let hB the singular hermitian metric on KX + L defined by hB |Xs := 1/Ks. We prove that hB has semipositive curvature. This is a generalization of the ...

Abstract In this article we show that for every finite area hyperbolic surface X of type {(g,n)} and any harmonic Beltrami differential ? on , then the magnitude at point small injectivity radius is uniform bounded from above by ratio Weil–Petersson norm over square root systole up to a positive constant multiplication. We apply bound Ricci curvature, restricted short in moduli space, uniformly...

Asymptotic models (small perturbation and small slope approximation at first-order, Kirchhoff approximation or two-scale model) used to predict the normalized radar cross section of the sea surface generally fail to reproduce in detail backscatter radar measurements. In particular, the predicted polarization ratio versus incidence and azimuth angles is not in agreement with experimental data. T...

PURPOSE To evaluate with the use of corneal topographic data the differences between total corneal power calculated using ray tracing (TCP) and the Gaussian formula (GEP) in normal eyes, eyes that previously underwent laser in situ keratomileusis/photorefractive keratectomy (LASIK/PRK), and theoretical models. METHODS TCP and GEP using mean instantaneous curvature were calculated over the cen...

Thickness stretching effect based on shear and normal deformation theory is used in this paper for magneto-electro-elastic vibration analysis of a three-layered curved nanobeam including a nano core and two piezo-magnetic layers. Size-dependency is included in derivation of governing equations of motion based Eringen's nonlocal elasticity theory. The initial curvature is accounted in calculatio...