Differential Geometry of Warped Product Manifolds and Submanifolds

The following sections are included:

• Dedication
• Foreword
• Preface
• Contents

Riemannian geometry was first put forward in generality by B. Riemann in the middle of nineteenth century. Riemannian geometry, including the Euclidean geometry and the classical non-Euclidean geometries as the most special particular cases, deals with a broad range of more general geometries whose metric properties vary from point to point.

Under the impetus of Einstein's Theory of General Relativity, the positiveness of the inner product induced from Riemannian metric was weakened to non-degeneracy. Consequently, one also has the notion of pseudo-Riemannian manifolds.

Development of Riemannian geometry resulted in a synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled Einstein's general relativity theory, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology. Since every manifold admits a Riemannian metric, Riemannian geometry often helps to solve problems of differential topology. Most remarkably, by applying Riemannian geometry, G. Y. Perelman proved Thurston's geometrization conjecture in 2003; consequently solved in the affirmative famous Poincaré's conjecture posed in 1904.

Differential geometry started during the seventeenth century as the study of curves in the Euclidean plane and of curves and surfaces in the Euclidean 3-space 𝔼³ by means of the techniques of differential calculus…

One of the most fruitful generalizations of the notion of Cartesian (or direct) products is the notion of warped products defined in [Bishop and O’Neill (1964)]. The concept of warped products appeared in the mathematical and physical literature before [Bishop and O’Neill (1964)]. For instance, warped products were called semi-reducible spaces in [Kruchkovich (1957)]…

Astronomers were unsure of the size of our galaxy at the beginning of the 20th century. Generally, they believed it was not much greater than a few tens of thousands of light years across. However, astronomers had noted fuzzy patches of light in the night sky, which are called nebulae. Some astronomers thought these could be distant galaxies…

Contact geometry studies a geometric structure on manifolds given by a hyperplane distribution in the tangent bundle and specified by a 1-form satisfying a maximum non-degeneracy condition. In view of the Frobenius theorem, one may recognize the condition as the opposite of the condition that the distribution be determined by a codimension one foliation on the manifold…

A Kähler manifold M is a 2n-dimensional manifold equipped with a complex structure J, J² = −I, and a compatible Riemannian metric g, namely,

Let N be a Riemannian submanifold of an almost Hermitian manifold $(\tilde{M},\tilde{g},J).$ For a given vector X ∈ TN, we put

A real submanifold N of an almost Hermitian manifold $\left(\tilde{M},\tilde{g},J\right)$ is called a generic submanifold if the maximal complex subspace, T_xN ∩ J(T_xN), of T_xN is of constant dimension along N. Generic submanifolds are “generic” in the sense that every real submanifold of an almost Hermitian manifold is the closure of the union of some generic submanifolds of the almost Hermitian manifold.

Some basic results on generic submanifolds of Kähler Manifolds can be found in [Chen (1981a,e)].

A generic submanifold of an almost Hermitian manifold $\left(\tilde{M},\tilde{g},J\right)$ is called a CR-submanifold if its purely real distribution ℌ^⊥ is a totally real distribution, i.e., $J{\mathscr{H}}_x^{\perp }\subset T_x^{\perp }N$ for each x ∈ N.

The notion of CR-submanifolds was introduced in [Bejancu (1978)]. Complex submanifolds and totally real submanifolds are special cases of CR-submanifolds. A CR-submanifold is called proper if it is neither a complex nor a totally real submanifold.

The main purpose of this chapter is to present basic results on proper CR-submanifolds.

According to the famous embedding theorem of J. F. Nash done in [Nash (1956)], every Riemannian manifold can be isometrically embedded in some Euclidean spaces with sufficiently high codimension. The Nash embedding theorem implies that every warped product N₁ ×_f N₂ can be isometrically embedded as Riemannian submanifolds in Euclidean spaces with sufficiently high codimension.

In this chapter we present many important results concerning the geometry of warped product manifolds immersed in Riemannian manifolds or in Kähler manifolds.

The following sections are included:

• Warped product CR-submanifolds
• CR-warped products and their characterization
• Examples of CR-warped products
• A general inequality for CR-warped products
• Twisted product CR-submanifolds
• Warped product submanifolds with a holomorphic factor
• Warped product hemi-slant submanifolds
• Warped product semi-slant submanifolds
• Warped product pointwise semi-slant submanifolds
• Warped product pointwise bi-slant submanifolds
• Warped products in locally conformal Kähler manifolds

The following sections are included:

• CR-warped products
• A PDE system associated with the basic equality
• CR-warped products in C^m satisfying basic equality
• CR-warped products in CP^m and CH^m
• CR-warped products with compact holomorphic factor

The following sections are included:

• Another optimal inequality for CR-warped products
• CR-warped products in C^m satisfying the equality
• CR-warped products in CP^m satisfying the equality
• CR-warped products in CH^m satisfying the equality
• Irreducibility of real hypersurfaces in non-flat complex space forms
• Warped product real hypersurfaces

Curvatures invariants play key roles in differential geometry as well as in physics. For instance, the magnitude of a force required to move an object at constant speed is, according to Newton’s law, a constant multiple of the curvature of the trajectory. The motion of a body in a gravitational field is determined by the curvatures of spacetime according to A. Einstein…

The following sections are included:

• Nearly Kähler manifolds
• Nearly Kähler structure on S⁶
• Complex submanifolds of nearly Kähler manifolds
• Lagrangian submanifolds of nearly Kähler manifolds
• CR-submanifolds in nearly Kähler manifolds
• Warped products in nearly Kähler manifolds
• Examples of warped product CR-submanifolds in nearly Kähler S⁶
• Non-existence of CR-products in nearly Kähler S⁶
• A special class of warped product submanifolds in nearly Kähler S⁶

Para-complex numbers were introduced as a generalization of complex numbers in [Graves (1845)]. J. T. Graves applied para-complex numbers to solve some questions in number theory.

Para-Kähler geometry is the geometry related to the algebra of paracomplex numbers. Properties of para-Kähler manifolds were first studied in [Rashevsky (1948)], in which P. K. Rashevsky considered a neutral metric of signature (n, n) defined from a potential function on a locally product 2n-manifold. He called such manifolds stratified space. Para-Kähler manifolds were explicitly defined in [Rozenfeld (1949)]. B. A. Rozenfeld compared Rashevsky's definition with Kähler's definition in the complex case and established the analogy between Kähler and para-Kähler manifolds.

The following sections are included:

• Sasakian manifolds and submanifolds
• Warped products in Sasakian manifolds
• Contact CR-submanifolds
• CR-warped products with smallest codimension
• Another inequality for contact CR-warped products in Sasakian manifolds
• Pointwise bi-slant and hemi-slant warped products in Sasakian manifolds

The following sections are included:

• Affine spaces and hypersurfaces
• Centroaffine hypersurfaces
• Graph hypersurfaces
• A realization problem for affine hypersurfaces
• Warped products as centroaffine hypersurfaces
• Warped products as graph hypersurfaces
• Realization of Robertson-Walker spaces as affine hypersurfaces

The following sections are included:

• Bibliography
• General Index
• Author Index