Local Isometric Immersions Research Articles

Local Isometric Immersions and Breakdown of Manifolds Determined by Cauchy Problems of the Degasperis–Procesi Equation

Local Isometric Immersions and Breakdown of Manifolds Determined by Cauchy Problems of the Degasperis–Procesi Equation

Characterization of framed curves arising from local isometric immersions

We characterize the class of framed curves that are induced by local isometric immersions (bending deformations) defined in a neighbourhood of a curve in the reference configuration. This characterization is sharp; in particular, for every framed curve belonging to the class, we construct a local isometric immersion from which it arises.

Clairaut surfaces in Euclidean three-space

In this note, we investigate local isometric immersion of Clairaut metrics in Euclidean three-space. A Clairaut metric is determined up to isometry by a single function of one variable. We show that an isometric immersion is formally determined by two functions of one variable, uniquely up to coordinate reflection and ambient Euclidean motions. Further, if the Clairaut metric and these two functions are real-analytic, there exists a local isometric immersion realizing these data. We give a more explicit description for a finite-dimensional family of examples.

Isometric immersions and differential equations describing pseudospherical surfaces

We provide families of second order nonlinear partial differential equations describing pseudospherical surfaces (pss equations) with the property of having local isometric immersions in E3 with principal curvatures depending on a jet of finite order of their solutions. These equations occupy a particularly special place amongst pss equations since a series of recent papers on several classes of pss equations seemed to suggest that only the sine-Gordon equation had the above property. Explicit examples of such equations are given, which include the short pulse equation and some generalizations.

A Novikov equation describing pseudo‐spherical surfaces, its pseudo‐potentials, and local isometric immersions

AbstractWe show that an equation discovered in V. Novikov [Generalizations of the Camassa–Holm equation, J Phys A: Math Theor. 2009; 42: paper 342002] describes pseudo‐spherical surfaces and is geometrically integrable. From the geometric structure of the equation we obtain an infinite hierarchy of conservation laws. The problem of local isometric immersions is also considered.

A One-Dimensional Singular Non-CSC Extremal Kähler Metric can be Isometrically Imbedded into $${\mathbb {R}}^3$$ as a Weingarten Surface

In this paper, we get a one-parameter family of local isometric immersions from a compact Riemann surface with a singular non-CSC extremal Kahler metric to $${\mathbb {R}}^3$$ , each of whom is a Weingarten surface. In fact, we can get explicit expressions of the mean curvatures in the family by the Gauss curvature of the metric.

Local isometric immersions of pseudo-spherical surfaces and kth order evolution equations

We consider the class of evolution equations of the form [Formula: see text], [Formula: see text], that describe pseudo-spherical surfaces. These were classified by Chern and Tenenblat in [Pseudospherical surfaces and evolution equations, Stud. Appl. Math 74 (1986) 55–83.]. This class of equations is characterized by the property that to each solution of such an equation, there corresponds a 2-dimensional Riemannian metric of constant curvature [Formula: see text]. Motivated by the special properties of the sine-Gordon equation, we investigate the following problem: given such a metric, is there a local isometric immersion in [Formula: see text] such that the coefficients of the second fundamental form of the immersed surface depend on a jet of finite order of [Formula: see text]? We extend our earlier results for second-order evolution equations [N. Kahouadji, N. Kamran and K. Tenenblat, Local isometric immersions of pseudo-spherical surfaces and evolution equations, Fields Inst. Commun. 75 (2015) 369–381; N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, Comm. Anal. Geom. 24(3) (2016) 605–643.] to [Formula: see text]th order equations by proving that there is only one type of equation that admit such an isometric immersion. More precisely, we prove under the condition of finite jet dependency that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions [Formula: see text] are universal, i.e. they are independent of [Formula: see text]. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the isometric immersion.

Local isometric immersions of pseudospherical surfaces described by evolution equations in conservation law form

For the pseudospherical surfaces described by a class of second order evolution equations, of the form zt=A(x,t,z)z2+B(x,t,z,z1), we consider the problem of local isometric immersion into the 3-dimensional Euclidean space E3 with a second fundamental form depending on finite-order jets of solutions z of the considered equations. We also provide an extension of our analysis to the case of k-th order evolution equations in conservation law form. Examples of equations admitting such local isometric immersions, are provided by equations like Burgers, Murray, Svinolupov–Sokolov, Kuramoto–Sivashinsky, Sawada–Kotera, Kaup–Kupershmidt, as well as hierarchies of evolution equations in conservation law form like Burgers, mKdV and KdV.

Third-order differential equations and local isometric immersions of pseudospherical surfaces

The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and nonlocal) and the presence of associated linear problems. It thus contains many important known examples of integrable equations, like the sine-Gordon, Liouville, KdV, mKdV, Camassa–Holm and Degasperis–Procesi equations, and is also home to many new families of integrable equations. Our paper is concerned with the question of the local isometric immersion in E3 of the pseudospherical surfaces defined by the solutions of equations belonging to the class introduced by Chern and Tenenblat [Pseudospherical surfaces and evolution equations, Stud. Appl. Math. 74 (1986) 55–83]. In the case of the sine-Gordon equation, it is a classical result that the second fundamental form of the immersion depends only on a jet of finite order of the solution of the partial differential equation. A natural question is therefore to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudospherical surfaces. In a pair of earlier papers [N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, to appear in Comm. Anal. Geom., arXiv: 1308.6545; Local isometric immersions of pseudo-spherical surfaces and evolution equations, in Hamiltonian Partial Differential Equations and Applications, eds. P. Guyenne, D. Nichols and C. Sulem, Fields Institute Communications, Vol. 75 (Springer-Verlag, 2015), pp. 369–381], it was shown that this property fails to hold for all [Formula: see text]th-order evolution equations [Formula: see text] and all other second-order equations of the form [Formula: see text], except for the sine-Gordon equation and a special class of equations for which the coefficients of the second fundamental form are universal, that is functions of [Formula: see text] and [Formula: see text] which are independent of the choice of solution [Formula: see text]. In this paper, we consider third-order equations of the form [Formula: see text], [Formula: see text], which describe pseudospherical surfaces with the Riemannian metric given in [T. Castro Silva and K. Tenenblat, Third order differential equations describing pseudospherical surfaces, J. Differential Equations 259 (2015) 4897–4923]. This class contains the Camassa–Holm and Degasperis–Procesi equations as special cases. We show that whenever there exists a local isometric immersion in E3 for which the coefficients of the second fundamental form depend on a jet of finite order of [Formula: see text], then these coefficients are universal in the sense of being independent on the choice of solution [Formula: see text]. This result further underscores the special place that the sine-Gordon equations seem to occupy amongst integrable partial differential equations in one space variable.

Second-order equations and local isometric immersions of pseudo-spherical surfaces

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 order zero, for all other differential equations, whenever such an immersion exists, the coefficients are universal functions of $x$ and $t$, independent of $u$.

A fundamental theorem for hypersurfaces in semi-Riemannian warped products

We give necessary and sufficient conditions for a semi-Riemannian manifold of arbitrary signature to be locally isometrically immersed into certain warped products. Then, we describe a way to use the structure equations of such immersions to construct foliations of marginally trapped surfaces in a four-dimensional Lorentzian spacetimes. We point out that, sometimes, Gauss and Codazzi equations are not sufficient to ensure the existence of a local isometric immersion of a semi-Riemannian manifold as a hypersurface of another manifold. We finally give two low-dimensional examples to illustrate our results.

On a Liu–Yau type inequality for surfaces

Let $\Omega$ be a compact and mean-convex domain with smooth boundary $\Sigma:=\partial\Omega$, in an initial data set $(M^3,g,K)$, which has no apparent horizon in its interior. If $\Sigma$ is spacelike in a spacetime $(\E^4,g\_\E)$ with spacelike mean curvature vector $\mathcal{H}$ such that $\Sigma$ admits an isometric and isospin immersion into $\mathbb{R}^3$ with mean curvature $H\_0$, then: \begin{eqnarray*} \int\_{\Sigma}|\mathcal{H}|d\Sigma\leq\int\_{\Sigma}\frac{H\_0^2}{|\mathcal{H}|}d\Sigma. \end{eqnarray*} If equality occurs, we prove that there exists a local isometric immersion of $\Omega$ in $\mathbb{R}^{3,1}$ (the Minkowski spacetime) with second fundamental form given by $K$. In Theorem liu-yau-minkowski, we also examine, under weaker conditions, the case where the spacetime is the $(n+2)$-dimensional Minkowski space $\mathbb{R}^{n+1,1}$ and establish a stronger rigidity result.

On real local isometric immersions of $${\mathbb{C}Q^2_c}$$ into $${\mathbb{C} P^{3}}$$

We investigate real local isometric immersions of Kahler manifolds \({\mathbb{C}Q^2_c}\) of constant holomorphic curvature 4c into complex projective 3-space. Our main result is that the standard embedding of \({\mathbb{C}P^2}\) into \({\mathbb{C}P^3}\) has strong rigidity under the class of local isometric transformations. We also prove that there are no local isometric immersions of \({\mathbb{C}Q^2_c}\) into \({\mathbb{C}P^3}\) when they have different holomorphic curvature. An important method used is a study of the relationship between the complex structure of any locally isometric immersed \({\mathbb{C}Q^2_c}\) and the complex structure of the ambient space \({\mathbb{C}P^3}\).

TOPOLOGICAL AND GEOMETRICAL PROPRIETIES OF BRANE-WORLDS

Through the characterization of a spherically symmetric space-time as a local brane-world immersed into six-dimensional pseudo-Euclidean spaces, with different signatures of the bulk, we investigate the existence of a topological difference in the immersed brane-world. In particular the Schwarzschild's brane-world and its Kruskal (or Frønsdal) brane-world extension are examined from point of view of the immersion formalism. We prove that there is a change of signature of the bulk when we consider a local isometric immersion and different topologies of a brane-world in that bulk.

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q c 4.

Isometric immersions into a homogeneous Lorentzian Heisenberg group and rigidity

AbstractIn this paper we prove an existence result for local and global isometric immersions of semi-Riemannian surfaces into the three dimensional Heisenberg group endowed with a homogeneous left–invariant Lorentzian metric. As a corollary, we prove a rigidity result for such immersions.

Lipschitz-continuous local isometric immersions: rigid maps and origami

A rigid map u : Ω ⊂ R n → R m is a Lipschitz-continuous map with the property that at every x ∈ Ω where u is differentiable then its gradient Du ( x ) is an orthogonal m × n matrix. If Ω is convex, then u is globally a short map, in the sense that | u ( x ) − u ( y ) | ⩽ | x − y | for every x , y ∈ Ω ; while locally, around any point of continuity of the gradient, u is an isometry. Our motivation to introduce Lipschitz-continuous local isometric immersions (versus maps of class C 1 ) is based on the possibility of solving Dirichlet problems; i.e., we can impose boundary conditions. We also propose an approach to the analytical theory of origami, the ancient Japanese art of paper folding. An origami is a piecewise C 1 rigid map u : Ω ⊂ R 2 → R 3 (plus a condition which exclude self intersections). If u ( Ω ) ⊂ R 2 we say that u is a flat origami. In this case (and in general when m = n ) we are able to describe the singular set Σ u of the gradient Du of a piecewise C 1 rigid map: it turns out to be the boundary of the union of convex disjoint polyhedra, and some facet and edge conditions ( Kawasaki condition) are satisfied. We show that these necessary conditions are also sufficient to recover a given singular set; i.e., we prove that every polyhedral set Σ which satisfies the Kawasaki condition is in fact the singular set Σ u of a map u, which is uniquely determined once we fix the value u ( x 0 ) ∈ R n and the gradient Du ( x 0 ) ∈ O ( n ) at a single point x 0 ∈ Ω ∖ Σ . We use this characterization to solve a class of Dirichlet problems associated to some partial differential systems of implicit type.

RIBAUCOUR TRANSFORMATIONS ON RIEMANNIAN SPACE FORMS IN LORENTZIAN SPACE FORM

We study Ribaucour transformations on nondegener- ate local isometric immersions of Riemannian space forms into Lorentzian space forms with ∞at normal bundles. They can be explained by dressing actions on the solution space of Lorentzian Grassmannian systems.

Isometric immersions of pseudo-Riemannian space forms

In this paper we study local isometric immersions f: M s n ( K)→ N s+ q 2 n−1 ( c) of a time-like n-submanifold M s n ( K) with constant sectional curvature K and index s into a pseudo-Riemannian space form N s+ q 2 n−1 ( c) with constant sectional curvature c and index s+ q, where q≥0, 1≤ s≤ n−1 and K≠ c. We first prove the existence of Chebyshev coordinates of a time-like submanifold M s n ( K) in certain conditions. Afterwards, we generalize the classical Bäcklund theorem for space-like (or time-like) submanifolds in N n−1 2 n−1 ( c) and N 1 2 n−1 ( c). Finally as an application, in the Chebyshev coordinates, we use the Bäcklund theorem to give a Bäcklund transformation and a permutability formula between the generalized sine-Laplace equation and the generalized sinh-Laplace equation.

Genuine Deformations of Submanifolds

We introduce the concept of genuine isometric deformation of an Euclidean submanifold and describe the geometric structure of the submanifolds that admit deformations of this kind. That an isometric deformation is genuine means that the submanifold is not included into a submanifold of larger dimension such that the deformation of the former is given by a deformation of the latter. Our main result says that an Euclidean submanifold together with a genuine deformation in low (but not necessarily equal) codimensions must be mutually ruled, and gives a sharp estimate for the dimension of the rulings. This has several strong local and global consequences. Moreover, the unifying character and geometric nature, as opposed to a purely algebraic one, of our result suggest that it should be the starting point for a deformation theory extending the classical Sbrana Cartan theory for hypersurfaces to higher codimensions. The isometric deformation problem for a given isometric immersion f : M → R of a Riemannian manifold into flat Euclidean space with codimension p and a positive integer q is to describe all possible isometric immersions f : M → R. A satisfactory answer to the local version of the problem for every hypersurface (p=1) and q = 1 going back almost a century is due to Sbrana [19] and Cartan [4]. However, basic questions, like the existence of Sbrana-Cartan hypersurfaces of the discrete type or the possibility of smoothly attaching different types of these deformable hypersurfaces, were answered positively only recently; see [10] and also [3] for a special case. The global version of the problem for hypersurfaces was solved in [11] and [18]. Nothing similar to the Sbrana-Cartan theory for codimensions q = p higher than one has yet been obtained. Nevertheless, the classical Beez-Killing rigidity theorem for hypersurfaces, the starting point for the theory, has several generalizations; see [1], [2], [5] and [20]. All these results provide generic algebraic conditions on the second fundamental form of the isometric immersion that imply isometric rigidity, that is, any other isometric immersion must differ by an isometry (rigid motion) of the ambient space. There is a large set of isometric deformations once we allow codimension q > p since one can always compose f : M → R with (local) isometric immersions of R into R. In [7] we found generic algebraic conditions on the second fundamental form