Academic literature on the topic 'Phi-curvature'

In this paper we study $\tau$-curvature tensor in $N(k)$-contact metric manifold. We study $\tau$-$\phi$-recurrent,$\tau$-$\phi$-symmetric and globally $\tau$-$\phi$-symmetric $N(k)$-contact metric manifold.

Let $(M,g(t))$ be a compact Riemannian manifold and the metric $g(t)$ evolve by the Ricci-Bourguignon flow. We find the formula variation of the eigenvalues of geometric operator $-\Delta_{\phi}+cR$ under the Ricci-Bourguignon flow, where $\Delta_{\phi}$ is the Witten-Laplacian operator and $R$ is the scalar curvature. In the final we show that some quantities dependent to the eigenvalues of the geometric operator are nondecreasing along the Ricci-Bourguignon flow on closed manifolds with nonnegative curvature.

Abstract In this paper, we study the eigenvalue problem of poly-drifting Laplacian on complete smooth metric measure space ( M , ⟨ , ⟩ , e − ϕ d v ) \left(M,\langle ,\rangle ,{e}^{-\phi }{\rm{d}}v) , with nonnegative weighted Ricci curvature Ric ϕ ≥ 0 {{\rm{Ric}}}^{\phi }\ge 0 for some ϕ ∈ C 2 ( M ) \phi \in {C}^{2}\left(M) , which is uniformly bounded from above, and successfully obtain several universal inequalities of this eigenvalue problem.

SUMMARY In this study, we devised a new set of analytical foundation solutions to compute the internal co-seismic displacement and strain changes caused by four independent point sources (strike-slip, dip-slip, horizontal tensile and vertical tensile) inside a homogeneous spherical earth model. Our model provides constraints on the deformation properties at depth and reveals that the internal co-seismic deformation is larger than that on the surface. The deformation near the source is convergent with our formulae. For the internal deformation at radial section plane, the patterns of horizontal displacements ${u_\theta },{u_\phi }$ and strain changes ${e_{{ rr}}},{e_{\theta \theta }},{e_{\phi \phi }},{e_{\theta \phi }}$ caused by strike-slip and tensile sources appear symmetric at the equidistance above and below the source. Their amplitudes are not identical but with a small discrepancy actually. Unlike these, the patterns of radial displacements ${u_r}$ for strike-slip and tensile sources exhibit point symmetry with the equidistance from the source. Also, the corresponding amplitudes are slightly different. The displacements ${u_\theta },{u_\phi }$ and strain changes ${e_{{ rr}}},{e_{\theta \theta }},{e_{\phi \phi }},{e_{\theta \phi }}$ caused by dip-slip also show the same properties as ${u_{ r}}$ of the strike-slip source. The magnitudes of the displacements and strain changes depend on the source types. The curvature effect on the near-field surface deformations is small, and it increases with the studied depth. However, for the far-field deformation caused by the strike-slip source (ds = 20 km), the curvature effect can be as large as 77 per cent when the epicentral distance approximates to 1778 km.

This paper deals with the study of geometry of Lorentzian para-Sasakian manifolds. We investigate some properties of D-conformally flat, D-conformally semi-symmetric, Xi-D-conformally flat and Phi-D-conformally flat curvature conditions on Lorentzian para-Sasakian manifolds. Also it is proved that in each curvature condition an LP-Sasakian manifold (Mn,g)(n>3) is an eta-Einstein manifold.Journal of Institute of Science and Technology, 2014, 19(1): 30-34

Abstract By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants $c_1(D)$ and $c_2(D)$ for a $d$-dimensional compact Riemannian manifold $D$ with boundary such that $c_1(D)\sqrt \lambda \|\phi \|_\infty \leqslant \|\nabla \phi \|_\infty \leqslant c_2(D)\sqrt \lambda \|\phi \|_\infty $ holds for any Dirichlet eigenfunction $\phi $ of $-\Delta $ with eigenvalue $\lambda $. In particular, when $D$ is convex with nonnegative Ricci curvature, the estimate holds for $c_1(D)= 1/{d\mathrm{e}}$ and $c_2(D)=\sqrt{\mathrm{e}}\left (\frac{\sqrt{2}}{\sqrt{\pi }}+\frac{\sqrt{\pi }}{4\sqrt{2}}\right ).$ Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper.

We have studied curvature symmetries in ($\epsilon$)-Kenmotsu manifolds. Next, we have proved the non-existence of a non-zero parallel 2-form in an ($\epsilon$)-Kenmotsu manifold. Moreover, we have characterised $\phi$-Ricci symmetric ($\epsilon$)-Kenmotsu manifolds and finally, we have proved that under certain restriction on the scalar curvature $divR$=0 and $divC$=0 are equivalent, where `$div$' denotes divergence.

A procedure for calculating left ventricular wall stresses segmentally was devised. Rectangular coordinates of the wall surfaces as seen in longitudinal section were plotted with the long axis as the x-axis. For each cavity point, a third-order polynomial (cubic spline) was fitted to the point together with several adjacent points on either side of it; the cavity radius (normal to cavity surface) at the point was found algebraically from the spline's coefficients. Each cavity radius was matched with the most symmetrical one from the opposite cavity surface. The point of intersection of the cavity radius with the outer surface was found, and a midwall point was identified from logarithmic means of cavity and outer radial lengths. For each midwall point, a cubic spline was fitted to that point together with several adjacent points on either side of it, and the midwall radius at that point was determined algebraically from the spline's coefficients. Each midwall radius was matched to the most symmetrical one from the opposite midwall. The locus of points at equal radial distances from opposite midwalls forms the axis. The midwall radius of curvature (r theta) orthogonal to the meridian at each point was taken as the radial distance from the midwall to the axis. Midwall meridional radius of curvature (r phi) was calculated from the spline's coefficients. Thickness (h) was calculated from intersections between the midwall radius and the inner and outer surfaces. For each point, meridional tension (T phi) was calculated as T phi = Pr theta/2, and hoop tension (T theta) was calculated as T theta = (Pr theta/2)(2 - r theta/r phi) where P is transmural pressure. Stresses were calculated as tensions divided by thicknesses (sigma phi = T phi/h, sigma theta = T theta/h), or more directly as sigma phi = Pr theta/2h and sigma theta = (Pr theta/2h)(2 - r theta/r phi). This procedure was validated with simple chamber shapes, and it has been applied to left ventricles.

AbstractThe Harry Dym hierarchy is derived with the help of Lenard recursion equations and zero curvature equation. Based on the Lax matrix, an algebraic curve $\mathcal{K}_{n}$ of arithmetic genus $n$ is introduced, from which the corresponding meromorphic function $\phi$ and Dubrovin-type equations are given. Further, the divisor and asymptotic properties of $\phi$ are studied. Finally, algebro-geometric solutions for the entire hierarchy are obtained according to above results and the theory of algebraic curve.

The aim of this thesis is to study the geometry of a Riemannian manifold M, with a special structure, called Einstein-type structure, depending on 3 real parameters, a smooth map phi into a target Riemannian manifold N, and a smooth function, called potential function, on M itself. We will occasionally let some of the parameters be smooth functions. The setting generalizes various previously studied situations:, Ricci solitons, almost Ricci-solitons, Ricci-harmonic solitons, quasi-Einstein manifolds and so on. By taking a constant potential function those structures reduces to harmonic-Einstein manifolds, that are a generalization of Einstein manifolds. The main ingredient of our analysis is the study of certain modified curvature tensors on M related to the map phi, called phi-curvatures, obtaining, for instance, their transformation laws under a conformal change of metric, and to develop a series of results for harmonic-Einstein manifolds that parallel those obtained for Einstein manifolds some times ago and also in the very recent literature. Einstein-type structures may be obtained, for some special values of the parameters involved, by a conformal deformation of a harmonic-Einstein manifold or even as the base of a warped product harmonic-Einstein manifold. The latter fact applies not only in the Riemannian but also in the Lorentzian setting and thus some Einstein-type structures are connected with solutions of the Einstein field equations, which are of particular interest in General Relativity. The main result of the thesis is the locally characterization, via a couple of integrability conditions and mild assumptions on the potential function, of Einstein-type structures with vanishing phi-Bach curvature (in the direction of the potential) as a warped product with harmonic-Einstein base and with an open real interval as fibre, extending in a very non trivial way a recent result for Bach flat Ricci solitons. Moreover the map phi depends only on the base of the warped product and not on the fibre . We also consider rigidity, triviality and non-existence results, both in the compact and non-compact cases. This is done via integral formulas and, in the non-compact case, via analytical tools, like the weak maximum principle and the classical results of Obata, Tashiro, Kanai.