Characterizing Base in Warped Product Submanifolds of Complex Projective Spaces by Differential Equations

In this study, a link between the squared norm of the second fundamental form and the Laplacian of the warping function for a warped product pointwise semi-slant submanifold Mn in a complex projective space is presented. Some characterizations of the base NT of Mn are offered as applications. We also look at whether the base NT is isometric to the Euclidean space Rp or the Euclidean sphere Sp, subject to some constraints on the second fundamental form and warping function.

Characterizing inequalities for existence of contact CR-warped product submanifolds of generalized Sasakian space forms admitting a nearly cosymplectic structure

This paper studies the contact CR-warped product submanifolds of a generalized Sasakian space form admitting a nearly cosymplectic structure. Some inequalities for the existence of these types of warped product submanifolds are established, the obtained inequalities generalize the results that have acquired in \cite{atceken14}. Moreover, we also estimate another inequality for the second fundamental form in the expressions of the warping function, this inequality also generalizes the inequalities that have obtained in \cite{ghefari19}. In addition, we also explore the equality cases.

On the Topology of Warped Product Pointwise Semi-Slant Submanifolds with Positive Curvature

In this paper, we obtain some topological characterizations for the warping function of a warped product pointwise semi-slant submanifold of the form Ωn=NTl×fNϕk in a complex projective space CP2m(4). Additionally, we will find certain restrictions on the warping function f, Dirichlet energy function E(f), and first non-zero eigenvalue λ1 to prove that stable l-currents do not exist and also that the homology groups have vanished in Ωn. As an application of the non-existence of the stable currents in Ωn, we show that the fundamental group π1(Ωn) is trivial and Ωn is simply connected under the same extrinsic conditions. Further, some similar conclusions are provided for CR-warped product submanifolds.

Leaps and bounds towards scale separation

In a broad class of gravity theories, the equations of motion for vacuum compactifications give a curvature bound on the Ricci tensor minus a multiple of the Hessian of the warping function. Using results in so-called Bakry-Émery geometry, we put rigorous general bounds on the KK scale in gravity compactifications in terms of the reduced Planck mass or the internal diameter. We reexamine in this light the local behavior in type IIA for the class of supersymmetric solutions most promising for scale separation. We find that the local O6-plane behavior cannot be smoothed out as in other local examples; it generically turns into a formal partially smeared O4.

Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold Ω p + q in the hyperbolic space ℍ m − 1 satisfy various extrinsic restrictions, then Ω p + q has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group π 1 Ω p + q is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.

Towards autonomous event identifications in wave-equation traveltime inversion

We present a method to automatically relate events between observed and synthetic data for wave-equation traveltime (WT) inversion. The scheme starts with local similarity measurements by applying cross-correlation to localized traces. We then develop a differentiable alternative of the argmax function. By applying the differentiable argmax to each time slice of the local similarity map, we build a traveltime difference function but keep this process differentiable. WT inversion requires only the traveltime difference of related events through a phase shift. Thus, we must reject events that are not apparently related between observed and synthetic data. The local similarity map implies the possibility of doing so but shows abrupt changes with time and offset. To mitigate this problem, we introduce a dynamic programming algorithm to define a warping function. Wave packets between observed and synthetic data are assumed to be related if they are connected by this warping function, and they also exhibit high local similarity. Only such related events are considered in the subsequent calculation of misfit and adjoint source. Numerical examples demonstrate that the proposed method successfully retrieves an informative model from roughly selected data. In contrast, WT inversion based on cross-correlation or deconvolution fails to do so.

On the geometry of submanifolds in certain warped products

In this paper, we deal with [Formula: see text]-dimensional submanifolds immersed in a slab of a warped product of the type [Formula: see text]. Under suitable constraints on the warping function [Formula: see text] and assuming that such a submanifold [Formula: see text] is either complete or stochastically complete, we apply some maximum principles in order to show that [Formula: see text] must be contained in a slice of [Formula: see text]. In particular, from our results we guarantee the nonexistence of [Formula: see text]-dimensional closed minimal submanifolds immersed in [Formula: see text]. Furthermore, we construct a nontrivial duo-graph in [Formula: see text] which illustrates the importance of our rigidity results.

Geometric Inequalities for Warped Products in Riemannian Manifolds

Warped products are the most natural and fruitful generalization of Riemannian products. Such products play very important roles in differential geometry and in general relativity. After Bishop and O’Neill’s 1969 article, there have been many works done on warped products from intrinsic point of view during the last fifty years. In contrast, the study of warped products from extrinsic point of view was initiated around the beginning of this century by the first author in a series of his articles. In particular, he established an optimal inequality for an isometric immersion of a warped product N1×fN2 into any Riemannian manifold Rm(c) of constant sectional curvature c which involves the Laplacian of the warping function f and the squared mean curvature H2 . Since then, the study of warped product submanifolds became an active research subject, and many papers have been published by various geometers. The purpose of this article is to provide a comprehensive survey on the study of warped product submanifolds which are closely related with this inequality, done during the last two decades.

Stability estimates for an inverse Steklov problem in a class of hollow spheres

In this paper, we study an inverse Steklov problem in a class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. Precisely, we aim at studying the continuous dependence of the warping function defining the warped product with respect to the Steklov spectrum. We first show that the knowledge of the Steklov spectrum up to an exponential decreasing error is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, when the warping functions are symmetric with respect to 1/2, we prove a log-type stability estimate in the inverse Steklov problem. As a last result, we prove a log-type stability estimate for the corresponding Calderón problem.