• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1285-1303
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• 2018

This note is motivated by gradient estimates of Li-Yau, Hamilton, and Souplet-Zhang for heat equations. In this paper, our aim is to investigate Yamabe equations and a non linear heat equation arising from gradient Ricci soliton. We will apply Bochner technique and maximal principle to derive gradient estimates of the general non-linear heat equation on Riemannian manifolds. As their consequence, we give several applications to study heat equation and Yamabe equation such as Harnack type inequalities, gradient estimates, Liouville type results.

• Kyungpook Mathematical Journal
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• v.58 no.2
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• pp.347-359
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• 2018

A Kenmotsu manifold $M^n({\phi},\;{\xi},\;{\eta},\;g)$, (n = 2m + 1 > 3) is called a generalized ${\phi}-recurrent$ if its curvature tensor R satisfies $${\phi}^2(({\nabla}_wR)(X,Y)Z)=A(W)R(X,Y)Z+B(W)G(X,Y)Z$$ for all $X,\;Y,\;Z,\;W{\in}{\chi}(M)$, where ${\nabla}$ denotes the operator of covariant differentiation with respect to the metric g, i.e. ${\nabla}$ is the Riemannian connection, A, B are non-vanishing 1-forms and G is given by G(X, Y)Z = g(Y, Z)X - g(X, Z)Y. In particular, if A = 0 = B then the manifold is called a ${\phi}-symmetric$. Now, a Kenmotsu manifold $M^n({\phi},\;{\xi},\;{\eta},\;g)$, (n = 2m + 1 > 3) is said to be generalized ${\phi}-Ricci$ recurrent if it satisfies $${\phi}^2(({\nabla}_wQ)(Y))=A(X)QY+B(X)Y$$ for any vector field $X,\;Y{\in}{\chi}(M)$, where Q is the Ricci operator, i.e., g(QX, Y) = S(X, Y) for all X, Y. In this paper, we study generalized ${\phi}-recurrent$ and generalized ${\phi}-Ricci$ recurrent Kenmotsu manifolds with respect to quarter-symmetric metric connection and obtain a necessary and sufficient condition of a generalized ${\phi}-recurrent$ Kenmotsu manifold with respect to quarter symmetric metric connection to be generalized Ricci recurrent Kenmotsu manifold with respect to quarter symmetric metric connection.

• The Pure and Applied Mathematics
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• v.9 no.2
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• pp.107-111
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• 2002

In this article, we prove various properties and some Liouville type theorems for quasi-strongly p-harmonic maps. We also describe conditions that quasi-strongly p-harmonic maps become p-harmonic maps. We prove that if $\phi$ : $M\;\longrightarrow\;N$ is a quasi-strongly p-harmonic map (\rho\; $\geq\;2$) from a complete noncompact Riemannian manifold M of nonnegative Ricci curvature into a Riemannian manifold N of non-positive sectional curvature such that the $(2\rho－2)$-energy, $E_{2p-2}(\phi)$ is finite, then $\phi$ is constant.

• Advances in aircraft and spacecraft science
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• v.4 no.6
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• pp.613-637
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• 2017

The development systems for the Structural Health Monitoring has attracted considerable interest from several engineering fields during the last decades and more specifically in the aerospace one. In fact, the introduction of those systems could allow the transition of the maintenance strategy from a scheduled basis to a condition-based approach providing cost benefits for the companies. The research presented in this paper consists of a definition and next comparison of four methods applied to numerical measurements for the extraction of damage features. The first method is based on the determination of the Structural Intensity field at the on-resonance condition in order to acquire information about the dissipation of vibrational energy throughout the structure. The Damage Quantification Indicator and the Average Integrated Global Amplitude Criterion methods need the evaluation of the Frequency Response Function for a healthy plate and a damaged one. The main difference between these two parameters is their mathematical definition and therefore the accuracy of the scalar values provided as output. The fourth and last method is based on the Mode-shape Curvature, a FRF-based technique which requires the application of particular finite-difference schemes for the derivation of the curvature of the plate. All the methods have been assessed for several damage conditions (the shape, the extension and the intensity of the damage) on two test plates: an isotropic (steel) plate and a 4-plies composite plate.

• Korean Journal of Mathematics
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• v.32 no.2
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• pp.315-328
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• 2024

The aim of the paper is to study an η-Einstein soliton on (2n + 1)-dimensional (k, µ)-contact metric manifold. At first, we establish various results related to (2n + 1)-dimensional (k, µ)-contact metric manifold that exhibit an η-Einstein soliton. Next we study some curvature conditions admitting an η-Einstein soliton on (2n+1)-dimensional (k, µ)-contact metric manifold. Furthermore, we consider specific conditions associated with an η-Einstein soliton on (2n+1)-dimensional (2n+1)-dimensional (k, µ)-contact metric manifold. Finally, we show the existance of an η-Einstein soliton on (k, µ)-contact metric manifold.

• Bulletin of the Korean Mathematical Society
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• v.42 no.2
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• pp.337-358
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• 2005

Let M be a real hypersurface with almost contact metric structure $(\phi,\;\xi,\;\eta,\;g)$ in a nonflat complex space form $M_n(c)$. In this paper, we prove that if the structure Jacobi operator $R_\xi$ commutes with both the structure tensor $\phi$ and the Ricc tensor S, then M is a Hopf hypersurface in $M_n(c)$ provided that the mean curvature of M is constant or $g(S\xi,\;\xi)$ is constant.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1219-1233
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• 2019

Let $E{\rightarrow}M$ be an arbitrary vector bundle of rank k over a Riemannian manifold M equipped with a fiber metric and a compatible connection $D^E$. R. Albuquerque constructed a general class of (two-weights) spherically symmetric metrics on E. In this paper, we give a characterization of locally symmetric spherically symmetric metrics on E in the case when $D^E$ is flat. We study also the Einstein property on E proving, among other results, that if $k{\geq}2$ and the base manifold is Einstein with positive constant scalar curvature, then there is a 1-parameter family of Einstein spherically symmetric metrics on E, which are not Ricci-flat.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1315-1325
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• 2021

In this paper, we firstly prove some general inequalities for the Neumann eigenvalues for domains contained in a Euclidean n-space ℝⁿ. Using the general inequalities, we can derive some new Neumann eigenvalues estimates which include an upper bound for the (k + 1)^th eigenvalue and a new estimate for the gap of the consecutive eigenvalues. Moreover, we give sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator with Navier boundary condition on compact Riemannian manifolds with boundary and positive Ricci curvature.