Geometry and Topology of Submanifolds IX

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• List of bibliographies 1962–1995

• Books on ADG and its applications

• Research articles, surveys, theses, and preprints

In this study we consider the contact metric Manifold M²ⁿ⁺¹(φ,ξ,η,g) with ξ belonging to the (κ,μ) nullity distribution. If ξ is parallel on χ(M) and (∇_Xh)(Y) = (∇_Yh)(X) then we show that M is a R-harmonic manifold. For a contact metric manifold M²ⁿ⁺¹ if g(X, φY) = {n(κ − 1) + 1)/κ}g(X, φhY) for every X, Y ∈ χ(M) where ξ belongs to the κ- nullity distribution (μ = 0), then we show that M is R-harmonic.

In this study we consider curves of type AW(k), 1 ≤ k ≤ 3. We give curvature conditions of these kind of curves. Finally we show that if γ is of type AW(k) then the cylinder built on γ is also of type AW(k).

We examine the centroaffine geometry of Tchebychev surfaces. By choosing local parameters adapted to the problem, it is possible to understand the integrability conditions. We introduce regular and singular surfaces and prove an existence theorem for regular ones. We will show that there are no Tchebychev surfaces with nonzero constant curvature metric, thus reducing the problem to K = 0, which has already been solved.

We study pairs of polar hypersurfaces in spheres. Like for polar pairs of hypersurfaces in Euclidean space we use a concept known from affine differential geometry, namely the notion of conjugate connections, as a basis for our investigation of polarization invariants.

We investigate hypersurfaces in space forms with some constant curvature functions. Examples of closed hypersurfaces in 𝕊⁴(1), in particular non-isoparametric ones, illustrate how subtle the situation might be. This motivates the local and global study of curvature pinching conditions in combination with constant curvature functions.

The following sections are included:

• Introduction
  • Plane curves
  • First inward cut-locus C
  • The elementary approach
  • Approach through the cut-locus
  • Approach through covering spaces
  • Approach through intersection properties

• Generalizations of these approaches
  • The elementary approach
  • The approach using intersections
  • The approach by covering spaces

• References

The following sections are included:

• Introduction

• The Lagrangian catenoid

• Characterizations

• References

The following sections are included:

• Introduction

• New Types of Riemannian Invariants

• δ-invariants and squared mean curvature

• A maximum principle and ideal immersions

• Relations between order of immersions and δ-invariants

• Relations between δ-invariants and λ₁

• New obstructions to minimal, Lagrangian and slant immersions

• Applications to rigidity problems

• Minimality of ideal Lagrangian immersions

• References

We describe curvature properties of several classes of manifolds of pseudosymmetric type, and discuss mutual relations among them.

The following sections are included:

• Conformally flat manifolds

• Conformally flat submanifolds

• Three dimensional conformally flat hypersurfaces

• References

The following sections are included:

• Introduction

• Preliminary results

• Hypersurfaces with two principal curvatures

• Hypersurfaces satisfying (3)

• References

The following sections are included:

• Introduction

• Numerical basics
  • One dimensional optimisation
    • Golden section search
    • Brent's method
  • Multi-dimensional optimisation
    • The Newton method
    • Quasi-Newton methods
    • Conjugate gradient methods
    • Choosing between conjugate gradient and quasi-Newton methods

• Classical fitting least-squares geometrical entities
  • Least-squares cylinder
  • Least-squares circle in 3D

• Re-parametrisation of the solution space
  • A surjective parametrisation
  • Minimising a function on submanifolds
    • Effects on the gradient
    • Effects on the Hessian
    • Conjugate directions
    • The role of the exponential map
  • Cylinders
  • Circles in 3D

• Comparative results and conclusion
  • Accuracy
  • Speed
  • Conclusion

• References

The following sections are included:

• Proposition

• Theorem

• References

The necessary and sufficient conditions of L*–invariance of n–fold integrals respectively to the one-parameter group of continuous transformations are obtained in terms of the Lie infinitesimal operators. The generalized Noether theorem is presented as a corollary.

Let Mⁿ be a Riemannian submanifold of codimension p in a real space form, with index p and constant curvature c. When p = 2, we obtain a pontwise inequality relating the normalized scalar curvature of Mⁿ and its main extrinsic invariants in , namely, the squared norm of the mean curvature vector H and the normalized scalar normal curvature. The case of equality is characterized, as in [DDW], in terms of the second fundamental form. Next, we study Riemannian surfaces of arbitrary codimension n in a semi-Riemannian real space form . We obtain an optimal inequality relating, besides c and ∥H∥², the Gauss curvature of the surface and the normal curvature. Using the notion of the ellipse of curvature, we obtain a characterization of the case of equality as in [GR]. In case of non-zero normal curvature tensor, we prove as in the Riemannian case [AFR] that the Euler characteristic of the 2-plane bundle containing the ellipse of curvature, is twice the Euler characteristic of the surface. Then, we give an example of rotation surfaces in .

We study third order Codazzi tensor fields relative to projectively flat connections. We construct such tensors from functions. This construction obeys a simple transformation law within a fixed projective class. As examples we consider cubic forms in affine hypersurface theory. In this case, our concept leads to third order PDEs which characterize affine spheres and quadrics.

We study higher order Codazzi tensors on constant curvature spaces and show how they can be generated by functions. We give applications in Riemannian geometry and hypersurface theory; in particular we characterize ellipsoids in terms of second order sherical harmonics.

In the present paper, we investigate some classes of submanifolds of a Sasakian manifold. B.-Y. Chen's inequalities for certain submanifolds are obtained. Classification theorems for some ideal such submanifolds are stated.

The following sections are included:

• Introduction

• Preliminaries

• Para-Hermitian structures on semidirect products of Lie groups

• Main results

• References

We study relations between Weyl geometries and Codazzi structures (see [1]) and investigate examples of Weyl geometries on affine hypersurfaces.

We extend a variational result of Blaschke and give some further stability results.

The following sections are included:

• Introduction

• Quaternions

• Quaternion functions and forms

• Generalisation of complex function theory

• Surface theory in quaternionic dress

• Harmonic maps

• Spin transformations

• Return to quaternions

• References