On the Fundamental Theorem of Submanifold Theory and Isometric Immersions with Supercritical Low Regularity

1. IntroductionThe main purpose of this paper is to introduce a new type of Riemannian curvature invariants and to show that these new invariants have interesting applications to several areas of mathematics; in particular, they provide new obstructions to minimal and Lagrangian isometric immersions. Moreover, these new invariants enable us to introduce and to study the notion of ideal immersions.One of the most fundamental problems in the theory of submanifolds is the immersibility (or non-immersibility) of a Riemannian manifold in a Euclidean space (or more generally, in a space form). According to a well-known theorem of J. F. Nash, every Riemannian manifold can be isometrically immersed in some Euclidean spaces with sufficiently high codimension.In order to study this fundamental problem, in view of Nash's theorem, it is natural to impose a suitable condition on the immersions. For instance, if one imposes the minimality condition on the immersions, it leads toPROBLEM 1. Given a Riemannian manifold M, what are the necessary conditions for M to admit a minimal isometric immersion in a Euclidean m-space Em?It is well-known that for a minimal submanifold in Em, the Ricci tensor satisfies Ric_??_0. For many years this was the only known general necessary Riemannian condition for a Riemannian manifold to admit a minimal isometric immersion in a Euclidean space.The main results of this article were presented at the 3rd Pacific Rim Geometry Conference held at Seoul, Korea in December 1996; also presented at the 922nd AMS meeting held at Detroit, Michigan in May 1997.

A basic question in submanifold theory is whether a given isometric immersion $f\colon M^n\to\R^{n+p}$ of a Riemannian manifold of dimension $n\geq 3$ into Euclidean space with low codimension $p$ admits, locally or globally, a genuine infinitesimal bending. That is, if there exists a genuine smooth variation of $f$ by immersions that are isometric up to the first order. Until now only the hypersurface case $p=1$ was well understood. We show that a strong necessary local condition to admit such a bending is the submanifold to be ruled and give a lower bound for the dimension of the rulings. In the global case, we describe the situation of compact submanifolds of dimension $n\geq 5$ in codimension $p=2$.

We prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric immersion in a Euclidean space via an eigenmap, then the eigenmap is a locally bi-Lipschitz embedding map to a sphere, which generalizes a fundamental theorem of Takahashi in submanifold theory to a non-smooth setting. Applications of these results include a topological sphere theorem and topological finiteness theorems, which are new even for closed Riemannian manifolds.

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold M endowed with the Webster metric hence consider a version of the CR Yamabe problem for CR manifolds with boundary. This occurs as the Yamabe problem for the Fefferman metric (a Lorentzian metric associated to a choice of contact structure θ on M , [20]) on the total space of the canonical circle bundle S 1 →C(M) → π M (a manifold with boundary ∂C(M)= π −1 (∂M) and is shown to be a nonlinear subelliptic problem of variational origin. For any real surface N={φ=0}⊂ H 1 we show that the mean curvature vector of N⇀ H 1 is expressed by H=− 1 2 ∑ j=1 2 X j (|Xφ | −1 X j φ)ξ provided that N is tangent to the characteristic direction T of ( H 1 , θ 0 ) , thus demonstrating the relationship between the classical theory of submanifolds in Riemannian manifolds (cf. e.g.[7]) and the newer investigations in [1], [6], [8] and [16]. Given an isometric immersion Ψ:N→ H n of a Riemannian manifold into the Heisenberg group we show that ΔΨ=2J T ⊥ hence start a Weierstrass representation theory for minimal surfaces in H n .

One of the basic problems in submanifold theory addressed in this book concerns the uniqueness of isometric immersions \(f\colon M^n\to \mathbb {Q}_c^m\) of Riemannian manifolds into space forms. Clearly, since g = τ ∘ f is also an isometric immersion for any isometry \(\tau \colon \mathbb {Q}_c^m\to \mathbb {Q}_c^m\), uniqueness should be understood to be up to congruences by isometries of the ambient space.

This paper concerns the submanifold geometry in the ambient space of warped product manifolds Fn ×σ ℝ, this is a large family of manifolds including the usual space forms ℝm, \( \mathbb{S}^m \) and ℝm. We give the fundamental theorem for isometric immersions of hypersurfaces into warped product space ℝn ×σ ℝ, which extends this kind of results from the space forms and several spaces recently considered by Daniel to the cases of infinitely many ambient spaces.

In this paper, by using new-concept pointwise bi-slant immersions, we derive a fundamental inequality theorem for the squared norm of the mean curvature via isometric warped-product pointwise bi-slant immersions into complex space forms, involving the constant holomorphic sectional curvature c, the Laplacian of the well-defined warping function, the squared norm of the warping function, and pointwise slant functions. Some applications are also given.

In this article we survey known results on geometric and topological obstructions to various important classes of isometric immersions in submanifold theory. In addition we present several related open problems. The main purpose of this article is an invitation for further research on the topics that it deals with.

Ricci curvature on warped product submanifolds in spheres with geometric applications

Here we develop a theory on the differential geometry of a lightlike hypersurface M of a proper semi-Riemannian manifold \(\bar M\) For this purpose, we introduce a non-degenerate screen distribution and construct the corresponding lightlike transversal vector bundle tr(TM) of M, consistent with the well-known theory of Riemannian submanifolds. This enables one to define the induced geometrical objects such as linear connection, second fundamental form, shape operator, etc., and to obtain the Gauss-Codazzi equations leading to the Fundamental Theorem of lightlike hyper-surfaces. It is noteworthy that the second fundamental form (and, therefore, the results on totally geodesic and totally umbilical lightlike hypersurfaces) is independent of the choice of a screen distribution.

In this chapter we establish several basic facts of the theory of submanifolds that will be used throughout the book. We first introduce the second fundamental form and normal connection of an isometric immersion by means of the Gauss and Weingarten formulas. Then we derive their compatibility conditions, namely, the Gauss, Codazzi and Ricci equations. The main result of the chapter is the Fundamental theorem of submanifolds, which asserts that these data are sufficient to determine uniquely a submanifold of a Riemannian manifold with constant sectional curvature, up to isometries of the ambient space. As an application, we classify totally geodesic and umbilical submanifolds of space forms. We introduce the relative nullity distribution as well as the notion of principal normal vector fields of an isometric immersion, and derive some of their elementary properties. Submanifolds with flat normal bundle are briefly discussed.

A fundamental theorem for hypersurfaces in semi-Riemannian warped products

We prove a Bonnet theorem for isometric immersions of submanifolds into the products of an arbitrary number of simply connected real space forms. Then we prove the existence of associate families of minimal surfaces in such products. Finally, in the case of 𝕊2 × 𝕊2, we give a complex version of the main theorem in terms of the two canonical complex structures of 𝕊2 × 𝕊2.

We prove the fundamental theorems for affine immersions into hyperquadrics (including affine spaces) with arbitrary codimension, which are generalizations of those for isometric immersions into space forms. As applications, the fundamental theorems for equiaffine immersions into hyperquadrics with arbitrary codimension are obtained.

This paper deals with the subject of infinitesimal variations of Euclidean submanifolds with arbitrary dimension and codimension. The main goal is to establish a Fundamental theorem for these geometric objects. Similar to the theory of isometric immersions in Euclidean space, we prove that a system of three equations for a certain pair of tensors are the integrability conditions for the differential equation that determines the infinitesimal variations. In addition, we give some rigidity results when the submanifold is intrinsically a Riemannian product of manifolds.