We prove the existence, uniqueness, and $W^{1,2}$-regularity for the solution to the Pfaff system with antisymmetric $L^2$-coefficient matrix in arbitrary dimensions. Hence, we establish the equivalence between the existence of $W^{2,2}$-isometric immersions and the weak solubility of the Gauss--Codazzi--Ricci equations on simply-connected domains. The regularity assumptions of these results are sharp. As an application, we deduce a weak compactness theorem for $W^{2,2}_{\rm loc}$-immersions.
We provide conditions under which an isometric immersion of a (warped) product of manifolds into a space form must be a (warped) product of isometric immersions.
Wextend the results obtained recently by G. D'Ambra and A. Loi towards the proof of a conjecture of M.Gromov on isometric immersions via non-free maps.
We investigate the behavior of the second fundamental form of an isometric immersion of a space form with negative curvature into a space form so that the extrinsic curvature is negative. If the immersion has flat normal bundle, we prove that its second fundamental form grows exponentially.
We $δ$-approximate strictly short (e.g. constant) maps between Riemannin manifolds $f_0:X^m\to Y^N$ for $N>>m^2/2$ by $C^\infty$-smooth isometric immersions $f_δ:X^m\to Y^N$ with curvatures $curv(f_δ) < \frac {\sqrt 3}δ$, for $δ\to 0$
In this paper, we show the rigidity of isometric immersions for a Riemannian manifold of dimension $n-1$ into the light cone of $n+1$ dimensional Minkowski, de Sitter and anti-de Sitter spacetimes for $n\geq 3$.
The aim of this short article is to investigate the possibility of existence of totally umbilical isometric immersions in R^3 with isothermal parametrization and harmonic metric.
Using the results of \cite{P1}, we get some estimates of warping functions for isometric immersions by changing the target manifolds by some types of Riemannian manifolds: constant space forms and Hermitian symmetric spaces. And we deal with equality cases and obtain their applications. Finally, we give some open problems.
We introduce a family of variational functionals for spinor fields on a compact Riemann surface $M$ that can be used to find close-to-conformal immersions of $M$ into $\mathbb{R}^3$ in a prescribed regular homotopy class. Numerical experiments indicate that, by taking suitable limits, minimization of these functionals can also yield piecewise smooth isometric immersions of a prescribed Riemannian metric on $M$.
In this paper, we parametrize the space of isometric immersions of the hyperbolic plane into the hyperbolic 3-space in terms of null-causal curves in the space of oriented geodesics. Moreover, we characterize "ideal cones" (i.e., cones whose vertices are on the ideal boundary) by behavior of their mean curvature.
In this paper, we study biharmonic isometric immersions of a surface into and biharmonic Riemannian submersion from 3-dimensional Berger spheres. We obtain a classification of proper biharmonic isometric immersions of a surface with constant mean curvature into Berger 3-spheres. We also give a complete classification of proper biharmonic Hopf tori in Berger 3-sphere. For Riemannian submersions, we prove that a Riemannian submersion from Berger 3-spheres into a surface is biharmonic if and only if it is harmonic.
A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifolds. We study a triharmonic isometric immersion into a space form of non-positively constant curvature. We show that if the domain is complete and both the 4-enegy and the L^4-norm of the tension field are finite, then such an immersion is minimal.
We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This result generalizes previous ones by M. do Carmo and E. Lima, as well as by M. do Carmo and F. Warner. It also settles affirmatively a conjecture by do Carmo and Warner. We establish a similar result for complete isometric immersions satisfying a stronger condition on the second funda…
In this paper, we give a definition of coherent tangent bundles of space form type, which is a generalized notion of space forms. Then, we classify their realizations in the sphere as a wave front, which is a generalization of a theorem of O'Neill and Stiel: any isometric immersion of the n-sphere into the (n+1)-sphere of the same sectional curvature is totally geodesic.
For a given simply connected Riemannian surface Sigma, we relate the problem of finding minimal isometric immersions of Sigma into S^2 x R or H^2 x R to a system of two partial differential equations on Sigma. We prove that a constant intrinsic curvature minimal surface in S^2 x R or H^2 x R is either totally geodesic or part of an associate surface of a certain limit of catenoids in H^2 x R. We also prove that if a non constant curvature Riemannian surface admits a continuous one-parameter family of minimal isometric immersions into S^2 x R or H^2 x R, then all these immersions are associate.
In this paper, we obtain a sufficient and necessary condition for a simply connected Riemannian manifold $(M^n, g)$ to be isometrically immersed, as a submanifold with codimension $p\geq 1$, into the product $S^k\times H^{n+p-k}$ of sphere and hyperboloid.
The number of constant mean curvature isometric immersions of a surface
在经典曲面理论中，只有很少的已知曲面例子允许非平凡等距变形，更少的是非单连通的。本文研究具有常中曲率H的定向非单连通曲面的浸入x：M \to R ^[3]的等距变形问题。我们证明了M的所有具有常数中曲率H的等距浸入的空间是R^3的模同余，有限或圆。当它是一个圆时，对于浸入x，M中的每个周期都有消失的力，当H不为0时，也有消失的扭矩。本文将极小曲面的刚性结果推广到常中曲率曲面。此外，我们确定封闭的向量值1-形式的周期给力和扭矩。
Several characterizations of umbilic points of submanifolds in arbitrary Riemannian and Lorentzian manifolds are given. As a consequence, we obtain new characterizations of spheres in the Euclidean space and of hyperbolic spaces in the Lorentz-Minkowski space. We also prove the Lorentzian version of a classical result by Cartan.
In this article, we study constant mean curvature isometric immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and we classify these isometric immersions when the surface has constant intrinsic curvature. As applications, we use the sister surface correspondence to classify the constant mean curvature surfaces with constant intrinsic curvature in the $3-$dimensional homogenous manifolds $\mathbb{E}(κ, τ)$ and we use the Torralbo-Urbano correspondence to classify the parallel mean curvature surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{H}^2 \time…
In this paper we investigate the existence of ``partially'' isometric immersions. These are maps f:M->R^q which, for a given Riemannian manifold M, are isometries on some sub-bundle H of TM. The concept of free maps, which is essential in the Nash--Gromov theory of isometric immersions, is replaced here by that of H-free maps, i.e. maps whose restriction to H is free. We prove, under suitable conditions on the dimension q of the Euclidean space, that H-free maps are generic and we provide, for the smallest possible value of q, explicit expressions for H-free maps in the following three setting…
We consider the class of evolution equations that describe pseudo-spherical surfaces of the form u\_t = F (u, $\partial$u/$\partial$x, ..., $\partial$^k u/$\partial$x^k), k $\ge$ 2 classified by Chern-Tenenblat. This class of equations is characterized by the property that to each solution of a differential equation within this class, there corresponds a 2-dimensional Riemannian metric of curvature-1. We investigate the following problem: given such a metric, is there a local isometric immersion in R 3 such that the coefficients of the second fundamental form of the surface depend on a jet of …
In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equations occupy a particularly special place amongst pss equations, since a series of recent papers [2, 7, 11, 12, 13], on several classes of pss equations, seemed to suggest that only the sine-Gordon equation had the above property. Explicit examples are given, which include the sh…
In this article, we study minimal isometric immersions of Kähler manifolds into product of two real space forms. We analyse the obstruction conditions to the existence of pluriharmonic isometric immersions of a Kähler manifold into those spaces and we prove that the only ones into $\mathbb{S}^{m-1}\times\mathbb{R}$ and $\mathbb{H}^{m-1}\times \mathbb{R}$ are the minimal isometric immersions of Riemannian surfaces. Futhermore, we show that the existence of a minimal isometric immersion of a Kähler manifold $M^{2n}$ into $\mathbb{S}^{m-1}\times\mathbb{R}$ and $\mathbb{S}^{m-k}\times \mathbb{H}^k…
A Riemannian manifold endowed with $k\ge2$ complementary pairwise orthogonal distributions is called a Riemannian almost $k$-product manifold. In the article, for the first time, we study the following problem: find a relationship between intrinsic and extrinsic invariants of a Riemannian almost $k$-product manifold isometrically immersed in another Riemannian manifold. For such immersions, we establish an optimal inequality that includes the mixed scalar curvature and the square of the mean curvature. Although Riemannian curvature tensor belongs to intrinsic geometry, a special part called th…
We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields on the manifold, satisfying a certain non-linear equation involving the Riemannian curvature tensor of $M$. Setting $k=1$, we proceed to recover the fundamental theorem of hypersurfaces. In the case of manifolds of positive sectional curvature and $n\geq 3$, we reduce the solvability of the Gauss and Codazzi equations to the cancelation of a set of obstructio…
On any compact Riemann surface there always exists a singular non-CSC (constant scalar curvature) extremal K$\ddot{a}$hler metric which is called a non-CSC HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper, by moving frames, we show that any non-CSC HCMU metric can not be isometrically minimal immersed into 3-dimensional real space forms even locally. In general, any non-CSC HCMU metric can not be isometrically immersed into 3-dimensional real space forms with constant mean curvature (CMC).
We consider the class of differential equations that describe pseudo-spherical surfaces of the form $u\_t=F(u,u\_x,u\_{xx})$ and $u\_{xt}=F(u, u\_x)$ given in Chern-Tenenblat \cite{ChernTenenblat} and Rabelo-Tenenblat \cite{RabeloTenenblat90}. We answer the following question: Given a pseudo-spherical surface determined by a solution $u$ of such an equation, do the coefficients of the second fundamental form of the local isometric immersion in $\mathbb{R}^3$ depend on a jet of finite order of $u$? We show that, except for the sine-Gordon equation, where the coefficients depend on a jet of orde…
We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove geometrical inequalities for a sub-Riemannian submanifold. As applications, inequalities are proved for submanifolds with mutually orthogonal distributions that include scalar and mutual curvature. For compact submanifolds, inequalities are obtained that are supported by known integral formulas for almost-product manifolds.
This article begins the theory of submanifolds into products of 2 or more space forms. The tensors $\mathbf{R}$, $\mathbf{S}$ and $\mathbf{T}$ defined by Lira, Tojeiro and Vitório at \cite{LTV} and the Bonnet theorem proved by them are generalized for the product of many space forms. Besides, some examples given by Mendonça and Tojeiro at \cite{MT} and the reduction of codimension theorem proved by them are also generalized.
Isometric immersions of constant curvature one metrics on the 2-sphere with two conical singularities into Euclidean 3-space: A partial solution to the Gálvez-Hauswirth-Mira problem
We introduce an elliptic regularization of the PDE system representing the isometric immersion of a surface in $\mathbb R^{3}$. The regularization is geometric, and has a natural variational interpretation.
We consider submanifolds into Riemannian manifold with metallic structures. We obtain some new results for hypersurfaces in these spaces and we express the fundamental theorem of submanifolds into products spaces in terms of metallic structures. Moreover, we define new structures called complex metallic structures. We show that these structures are linked with complex structures. Then, we consider submanifolds into Riemannian manifold with such structures with a focus on invariant submanifolds and hypersurfaces. We also express in particular the fundamental theorem of submanifolds of complex s…
We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq p\leq \infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact $2$-Riem…
Isometric immersions into 3-dimensional homogeneous manifolds
给出了二维黎曼流形局部等距浸入具有四维等距群的三维齐次流形的充要条件。该条件表示的度量，第二基本型，并从周围的Killing场所产生的数据。这类三维流形特别包括Berger球面、Heisenberg空间Nil（3）、李群PSL（2，R）的泛覆盖以及积空间S^2 x R和H^2 x R。我们给出了这些流形中的常中曲率（CMC）曲面的一些应用;特别地，我们证明了广义Lawson对应的存在性，即，齐次三维流形中CMC曲面之间的局部等距对应。
We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold endowed with a polar metric. For a product manifold endowed with a polar metric, our main result gives a complete description of all its isometric immersions into a space form whose second fundamental forms are adapetd to its product structure, in the sense that the tangent spaces to each factor are preserved by all shape operators. This is a far-reaching general…
We investigate isometric immersions of locally conformally Kaehler metrics into Hopf manifolds. In particular, we study Hopf-induced metrics on compact complex surfaces.
Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space $\mathbb{R}^n$ in terms of an integral representation of Weierstrass type. Restricting to the case of immersions in $\mathbb{R}^4$, we study harmonicity conditions for such immersions and give a complete classification of CR-pluriharmonic immersions.
Although the Nash theorem solves the isometric embedding problem, matters are inherently more involved if one is further seeking an embedding that is well-behaved from the standpoint of submanifold geometry. More generally, consider a Lipschitz map $F:M^m\to\mathbb R^n$, where $M^m$ is a Hadamard manifold whose curvature lies between negative constants. The main result of this paper is that $F$ must perform a substantial compression: For every $r>0$ and integer $k\geq 2$ there exist $k$ geodesic balls of radius $r$ in $M^m$ that are arbitrarily far from each other, but whose images under $F$ a…
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 m…
We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugated two-dimensional immersions embedded into $\mathbb{R}^3$. The main tool we use is the relative entropy method employed in the setting of differential geometry for the Gauss-Codazzi system. The result allows us to compare also two solutions to the Gauss-Codazzi system that correspond to a smooth and a $C^{1,1}$ isometric immersion of not necessarily the same metric and prove continuous dependence of their second fundamental forms in terms of the metric and initial data in $L^2$.
The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is considered in this paper through the Gauss-Codazzi system for the second fundamental forms. It is shown that if the Gauss curvature satisfies an integrability condition, the surface has a global smooth isometric immersion in the three-dimensional Euclidean space even if the Gauss curvature decays very slowly at infinity. The new idea of the proof is based o…
In this paper the method of compensated compactness is applied to the problem of isometric immersion of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Previous applications of the method to this problem have required decay of order $t^{-4}$ in the Gauss curvature. Here we show that the decay of Hong $t^{-2-δ/2}$ where $δ\in(0,4)$ suffices.
The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip and disk geometries. By finding explicit piecewise smooth solutions of hyperbolic Monge-Ampere equations on a strip, we show there exist periodic isometric immersions of hyperbolic surfaces in the small slope regime. We extend these solutions to exact isometric immersions through …
We prove the developability and $C^{1,1/2}$ regularity of $W^{2,2}$ isometric immersions of $n$-dimensional domains into $R^{n+1}$. As a conclusion we show that any such Sobolev isometry can be approximated by smooth isometries in the $W^{2,2}$ strong norm, provided the domain is $C^1$ and convex. Both results fail to be true if the Sobolev regularity is weaker than $W^{2,2}$.
