Total Mean Curvature and Submanifolds of Finite Type

The following sections are included:

• Dedication
• Foreword
• Preface
• Contents

A differentiable manifold is a type of manifold that is locally similar enough to a Euclidean space to allow one to do calculus. Any manifold can be described by a collection of charts. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart…

Riemannian geometry was first put forward in generality by B. Riemann in the middle of nineteenth century. In his famous inaugural lecture “Über die Hypothesen welche der Geometrie zu Grunde liegen”, at Göttingen, Riemann discussed the foundations of geometry, introduced n-dimensional manifolds, formulated the concept of Riemannian manifolds and defined their curvature [Riemann (1854)]. Riemannian geometry, including the Euclidean geometry and the classical non-Euclidean geometries as most special particular cases, deals with a broad range of more general geometries whose metric properties vary from point to point…

Hodge theory is a branch of algebraic geometry, algebraic topology and complex manifold theory that deals with the decomposition of the cohomology groups of a complex projective algebraic variety. The theory was developed in the 1930s by W. V. D. Hodge (1903–1975) as an extension of de Rham cohomology. Hodge theory has major applications to Riemannian manifolds, Kähler manifolds and algebraic geometry…

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…

The following sections are included:

• Introduction
• Total absolute curvature of Chern and Lashof
• Willmore's conjecture and Marques-Neves' theorem
• Total mean curvature and conformal invariants
• Total mean curvature for arbitrary submanifolds
• A variational problem on total mean curvature
• Surfaces in 𝔼^m which are conformally equivalent to flat surfaces
• Total mean curvatures for surfaces in 𝔼⁴
• Normal curvature and total mean curvature of surfaces

The following sections are included:

• Introduction
• Order and type of submanifolds and maps
• Minimal polynomial criterion
• A variational minimal principle
• Finite type immersions of homogeneous spaces
• Curves of finite type
• Classification of 1-type submanifolds
• Submanifolds of finite type in Euclidean space
• 2-type spherical hypersurfaces
• Spherical k-type hypersurfaces with k ≥ 3
• Finite type hypersurfaces in hyperbolic space
• 2-type spherical surfaces of higher codimension

A pseudo-Riemannian submanifold M of a pseudo-Euclidean m-space is called biharmonic if its position vector field x satisfies

The following sections are included:

• (k, ℓ, λ)-harmonic maps and submanifolds
• Null 2-type hypersurfaces
• Null 2-type submanifolds with parallel mean curvature
• Null 2-type submanifolds with constant mean curvature
• Marginally trapped null 2-type submanifolds
• λ-biharmonic submanifolds of
• λ-biharmonic submanifolds in H^m
• λ-biharmonic submanifolds in S^m and

In this chapter we present numerous applications of finite type theory to several important subjects in differential geometry for Riemannian submanifolds and maps via their spectral decompositions.

The following sections are included:

• Pointwise finite type maps
• Submanifolds with finite type Gauss map
• Submanifolds with pointwise 1-type Gauss map
• Submanifolds with finite type spherical Gauss map
• Finite type submanifolds in Sasakian manifolds
• Legendre submanifolds satisfying ΔH_ϕ = λH_ϕ
• Geometry of tensor product immersions
• Finite type quadric and cubic representations
• Finite type submanifolds of complex projective space
• Finite type submanifolds of complex hyperbolic space
• Finite type submanifolds of real hyperbolic space
• L_r finite type hypersurfaces

The following sections are included:

• Bibliography
• Subject Index
• Author Index