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Register a new userAn almost splitting theorem for a warped product space
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Authors
Paul Woon Yin Lee
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We prove an almost splitting theorem for the warped product space with warped function $f(r)=coshleft(rsqrt{frac{lambda}{n-2}}right)$.
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In this paper, we strengthen the splitting theorem proved in [14, 15] and provide a different approach using ideas from the weak KAM theory.
We give a characterization {sl `a la Obata} for certain families of Kahler manifolds. These results are in the same line as other extensions of the well-known Obatas rigidity theorem from cite{Obata62}, like for instance the generalizations in cite{RanjSant97} and cite{Santhanam07}. Moreover, we give a complete description of the so-called Kahler doubly-warped product structures whose underlying metric is Einstein.
We give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type $N^n times_f Q^q$, where $f in C^infty(N)$. We also study the essential spectrum of these minimal submanifolds.
We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequalities involving the Laplacian of the warping function and the squared mean curvature of $C$-totally real warped product submanifolds of $(kappa ,mu ) $-space forms, Sasakian space forms and non-Sasakian $(kappa ,mu) $-manifolds. Then we obtain obstructions to the existence of minimal isometric immersions of $C$-totally real warped product submanifolds in $(kappa ,mu) $-space forms, non-Sasakian $(kappa ,mu) $-manifolds and Sasakian space forms. In the last, we obtain an example of a warped product $C$-totally real submanifold of a non-Sasakian $(kappa ,mu) $-manifold, which satisfies the equality case of the basic inequality.
We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold endowed with a polar metric. For a product manifold endowed with a polar metric, our main result gives a complete description of all its isometric immersions into a space form whose second fundamental forms are adapetd to its product structure, in the sense that the tangent spaces to each factor are preserved by all shape operators. This is a far-reaching generalization of a basic decomposition theorem for isometric immersions of Riemannian products due to Moore as well as its extension by Nolker to isometric immersions of warped products.
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