Abstract
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.
Highlights
Introduction and Main ResultsA classical challenge in Riemannian geometry is to discuss the geometrical and topological structures of submanifolds
The stable currents and homology groups are the most important characterizations of the Riemannian submanifolds because they control the behavior of the topology of submanifolds
The notion of non-existence stable current and vanishing homology on pinching the second fundamental form was introduced by Lawson-Simons [1]
Summary
A classical challenge in Riemannian geometry is to discuss the geometrical and topological structures of submanifolds. We deduce the following result from Theorem 2 and Remark 1 for the nonexistence of stable integrable l-currents and homology groups in the CR-warped product submanifolds of the complex projective space CP2m(4). (b) The i integral homology groups of Ωn with integer coefficients vanished; that is, Hl(Ωn, G) = Hk(Ωn, G) = 0, Using the result of Theorem 3, we can recall the sphere theorem for the compact oriented CR-warped product submanifold of a complex projective space CP2m(4) due to Chen [22], that is, Corollary 2. Let Ωn = NTl × f Nφk be compact, oriented warped product pointwise semi-slant submanifolds of the complex projective space CP2m(4); that is, f is a non-constant eigenfunction of the first non-zero eigenvalue λ1. Because this paper is connected to both warped product manifold and homotopy-homology theory, its results can be used as physical applications
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