Theory Of Immersions Research Articles

Infinitesimal variations of submanifolds

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.

Infinitesimal Variations of Submanifolds

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.

A class of complete minimal submanifolds and their associated families of genuine deformations

Concerning the problem of classifying complete submanifolds of Euclidean space with codimension two admitting genuine isometric deformations, we believe that the only known examples with the maximal possible rank four are the real Kaehler minimal submanifolds classified in parametric form by Dajczer-Gromoll [11]. They behave like minimal surfaces, namely, if simple connected they are either holomorphic or they admit a nontrivial one-parameter family of isometric deformations. Here we characterize a new class of complete genuinely deformable Euclidean submanifolds of rank four but now the structure of their second fundamental and the way it gets modified while deforming is quite more involved than in the Kaehler case. In particular, we see this as a strong indication that the classification problem is quite challenging. Being minimal, the submanifolds we introduced are also interesting by themselves. Some of the very basic question in the local and global theory of isometric immersions of Riemannian manifolds into Euclidean space remain in good part unanswered. For instance, outside some special cases it is not known which is the lowest codimension for which a given Riemannian manifold admits an isometric immersion. On one hand, there are several results that assure that a submanifold must be unique, that is, isometrically rigid, when lying in its lowest possible codimension. On the other hand, there are only a few theorems classifying isometrically deformable submanifolds and their deformations. This is due to the fact that rigidity is a “generic” property while being deformable is certainly not, and hence a situation harder to deal with. The exception for the deformation problem is the case of hypersurfaces. In fact, in the local case the problem was mostly solved by Sbrana [18] and Cartan [1] about a century ago; see [7] for details and a modern presentation. A solution to the problem for compact hypersurfaces was given by Sacksteder [17] and by Dajczer-Gromoll [10] in the complete case. But solving the deformation problem in codimension two turned out to be very challenging even in the more restrictive case of complete manifolds. In dealing with the isometric deformation problem in higher codimension, it has to be taken into account that any submanifold of a deformable submanifold has the isometric

Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for $\mathbb{P}^1_{a,b,c}$

This paper gives a new way of constructing Landau–Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger –Yau–Zaslow and Fukaya–Oh–Ohta–Ono. Moreover, we construct a canonical functor from the Fukaya category to the mirror category of matrix factorizations. This functor derives homological mirror symmetry under some explicit assumptions. As an application, the construction is applied to spheres with three orbifold points to produce their quantum-corrected mirrors and derive homological mirror symmetry. Furthermore, we discover an enumerative meaning of the (inverse) mirror map for elliptic curve quotients.

Integrable embeddings and foliations

A $k$-submanifold $L$ of an open $n$-manifold $M$ is called {\it weakly integrable (WI)} [resp. {\it strongly integrable (SI)}] if there exists a submersion $\Phi:M\rightarrow {\Bbb{E}}^{n-k}$ such that $L\subset \Phi^{-1}(0)$ [resp. $L= \Phi^{-1}(0)$]. In this work we study the following problem, first stated in a particular case by Costa et al. ({\it Invent. Math.} 1988): which submanifolds $L$ of an open manifold $M$ are WI or SI? For general $M$, we explicitly solve the case $k=n-1$ and provide necessary and sufficient conditions for submanifolds to be WI and SI in higher codimension. As particular cases we recover the theorem of Bouma and Hector ({\it Indagationes Math.} 1983) asserting that any open orientable surface is SI in ${\Bbb{E}}^3$, and Watanabe's and Miyoshi's theorems ({\it Topology} 1993 and 1995) claiming that any link is WI in an open $3$-manifold. In the case $M={\Bbb{E}}^n$ we fully characterize WI and SI submanifolds, we provide examples of $3$- and $7$-manifolds which are not WI and we show that a theorem by Miyoshi ({\it Topology} 1995) which states that any link in ${\Bbb{E}}^3$ is SI does not hold in general. The right analogue to Miyoshi's theorem is also proved, implying in particular the surprising result that no knot in ${\Bbb{E}}^3$ is SI. Our results applied to the theory of foliations of Euclidean spaces give rise to some striking corollaries: using some topological invariants we classify all the submanifolds of ${\Bbb{E}}^n$ which can be realized as proper leaves of foliations; we prove that ${\Bbb{S}}^3$ can be realized as a leaf of a foliation of ${\Bbb{E}}^n$, $n \geq 7$, but not in ${\Bbb{E}}^5$ or ${\Bbb{E}}^6$, which partially answers a question by Vogt ({\it Math. Ann.} 1993); we construct open $3$-manifolds which cannot be leaves of a foliation of any compact $4$-manifold but are proper leaves in ${\Bbb{E}}^4$. The theory of WI and SI submanifolds is a framework where many classical tools of differential and algebraic topology play a prominent role: Phillips-Gromov h-principle, Hirsh-Smale theory, complete intersections, Seifert manifolds, the theory of immersions and embeddings, obstruction theory, etc.

Generalized pseudohermitian manifolds

We define a generalized pseudohermitian structure on an almost CR manifold as a pair (h,P), where h is a positive definite fiber metric h on compatible with J, and is a smooth projector such that . We show that to each generalized pseudohermitian structure one can associate a canonical linear connection on the holomorphic bundle which is invariant under equivalence. This fact allows us to solve the equivalence problem in the case where is a kind 2 distribution. We study the curvature of the canonical connection, especially for the classes of standard homogeneous manifolds and 3-Sasakian manifolds. The basic formulas for isopseudohermitian immersions are also obtained in the attempt to enlarge the theory of pseudohermitian immersions between strongly pseudoconvex pseudohermitian manifolds of hypersurface type.

Partially isometric immersions and free maps

In this paper we investigate the existence of “partially” isometric immersions. These are maps \({f:M\rightarrow \mathbb{R}^q}\) which, for a given Riemannian manifold M, are isometries on some sub-bundle \({\mathcal{H}\subset TM}\). The concept of free maps, which is essential in the Nash–Gromov theory of isometric immersions, is replaced here by that of \({\mathcal{H}}\) –free maps, i.e. maps whose restriction to \({\mathcal{H}}\) is free. We prove, under suitable conditions on the dimension q of the Euclidean space, that \({\mathcal{H}}\) –free maps are generic and we provide, for the smallest possible value of q, explicit expressions for \({\mathcal{H}}\) –free maps in the following three settings: 1–dimensional distributions in \({\mathbb{R}^2}\), Lagrangian distributions of completely integrable systems, Hamiltonian distributions of a particular kind of Poisson Bracket.

Singular Riemannian foliations on nonpositively curved manifolds

We prove the nonexistence of a proper singular Riemannian foliation admitting section in compact manifolds of nonpositive curvature. Then we give a global description of proper singular Riemannian foliations admitting sections on Hadamard manifolds. In addition by using the theory of taut immersions we provide a short proof of this result in the special case of a polar action.

PARTIAL ACTIONS OF GROUPS

A partial action of a group G on a set X is a weakening of the usual notion of a group action: the function G×X→X that defines a group action is replaced by a partial function; in addition, the existence of g·(h·x) implies the existence of (gh)·x, but not necessarily conversely. Such partial actions are extremely widespread in mathematics, and the main aim of this paper is to prove two basic results concerning them. First, we obtain an explicit description of Exel's universal inverse semigroup [Formula: see text], which has the property that partial actions of the group G give rise to actions of the inverse semigroup [Formula: see text]. We apply this result to the theory of graph immersions. Second, we prove that each partial group action is the restriction of a universal global group action. We describe some applications of this result to group theory and the theory of E-unitary inverse semigroups.

Helgason spheres of compact symmetric spaces and immersions of finite type

A unit speed curve γ = γ(s) in a Riemannian manifold N is called a circle if there exists a unit vector field Y(s) along γ and a positive constant k such that ∇sγ′(s) = kY(s), ∇sY(s) = −kγ′(s). A maximal totally geodesic sphere with maximal sectional curvature in a compact irreducible symmetric space M is called a Helgason sphere. A circle which lies in a Helgason sphere of a compact symmetric space is called a Helgason circle. In this article we establish some fundamental relationships between Helgason circles, Helgason spheres of irreducible symmetric spaces of compact type and the theory of immersions of finite type.

Locally symmetric immersions

We use reflections with respect to submanifolds and related geometric results to develop, inspired by the work of Ferus and other authors, in a unified way a local theory of extrinsic symmetric immersions and submanifolds in a general analytic Riemannian manifold and in locally symmetric spaces. In particular we treat the case of real and complex space forms and study additional relations with holomorphic and symplectic reflections when the ambient space is almost Hermitian. The global case is also taken into consideration and several examples are given.

Affine immersion ofn-dimensional manifold intoR n+n(n+1)/2 and affine minimality

We formulate an affine theory of immersions of ann-dimensional manifold into the Euclidean space of dimensionn+n(n+1)/2 and give a characterization of critical immersions relative to the induced volume functional in terms of the affine shape operator.

SMOOTH IMMERSIONS OF MANIFOLDS OF LOW DIMENSION

A method is developed for solving a certain class of topological problems, arising in the theory of immersions of smooth manifolds, by studying the local multisingularities of differentiable mappings in general position. Bibliography: 20 titles.

The parallel rank of a submanifold of Euclidean space

The aim of this paper is to make some remarks about the theory of parallel immersions in ℝn. We show that the existence of parallels to a curve in ℝndepends on the values of the integrals of then− 1 curvature functions along the curve. We then give examples of immersions ofS1in ℝ3and of immersions of codimension 2 with local parallel rank zero, thereby answering certain questions in [1].

Higher-order nondegenerate immersions of manifolds

In this paper we show how the Smale-Hirsch theory of immersions can be adapted to get a simple proof of an integrability theorem of Gromov and Eliashberg concerning higher-order nondegenerate immersions.

Surgery on codimension one immersions in 𝑅ⁿ⁺¹: removing 𝑛-tuple points

The self-intersection sets of immersed n n -manifolds in ( n + 1 ) (n + 1) -space provide invariants of the n n th stable stem and the ( n + 1 ) (n + 1) st stable homotopy of infinite real projective space. Theorems of Eccles [5] and others [1, 8, 14, 19] relate these invariants to classically defined homotopy theoretic invariants. In this paper a surgery theory of immersions is developed; the given surgeries affect the self-intersection sets in specific ways. Using such operations a given immersion may be surgered to remove ( n + 1 ) (n + 1) -tuple and n n -tuple points, provided the Z / 2 {\mathbf {Z}}/2 -valued ( n + 1 ) (n + 1) -tuple point invariant vanishes ( n ≥ 5 ) (n \geq 5) . This invariant agrees with the Kervaire invariant for n = 4 k + 1 n = 4k + 1 . These results first appeared in my dissertation [2]; a summary was presented in [3]. Some results and methods have been improved since these works were written. In particular, the proof of Theorem 14 has been simplified.

ISOMETRIC IMMERSIONS OF DOMAINS OF THREE-DIMENSIONAL LOBACHEVSKY SPACE IN FIVE-DIMENSIONAL EUCLIDEAN SPACE, AND THE MOTION OF A RIGID BODY

In this paper a connection is established between the theory of isometric immersions of a space of constant curvature and a classical problem of mechanics on the motion of a rigid body.Figures: 3. Bibliography: 6 titles.

Helical minimal immersions of compact Riemannian manifolds into a unit sphere

An isometric immersion of a Riemannian manifold M into a Riemannian manifold M is called helical if the image of each geodesic has constant curvatures which are independent of the choice of the particular geodesic. Suppose M is a compact Riemannian manifold which admits a minimal helical immersion of order 4 into the unit sphere. If the Weinstein integer of M equals that of one of the projective spaces, then M is isometric to that projective space with its canonical metric. 0. Introduction. In [2], Besse constructed a minimal immersion with a nice property of a strongly harmonic manifold into a sphere. This nice property is that the images of geodesics of the strongly harmonic manifold are of constant curvatures as curves in the sphere, and the curvatures and the osculating orders are independent of geodesics. Immersions with such a property are said to be helical (cf. [8]). Sakamoto [8] studied a helical immersion of a Riemannian manifold M into a unit sphere. The result is that if M is compact, then it is a Blaschke manifold and, moreover, if the helical immersion is minimal, then M is a globally harmonic manifold. It is well known that if a globally harmonic manifold is compact and simply connected, then it is a strongly harmonic manifold (Michel's theorem, cf. [2]). Thus, we can declare that the theory of helical minimal immersions of compact simply connected Riemannian manifolds into a unit sphere is a submanifold version of strongly harmonic manifold theory. Let f: M S(1) be a helical immersion of a compact Riemannian manifold M into a unit sphere S(1). Since M is a Blaschke manifold, all geodesics of M are simply closed and of the same length. Furthermore, if we denote the cut-locus of x E M by Cut(x), then the unit tangent vectors at x of geodesics emanating from x and entering to y E Cut(x) compose a great sphere in the unit tangent sphere at x. The dimension of the great sphere is independent of x and it is equal to 0, 1, 3, 7, or n 1 (n = dim M), which is the index of the first conjugate point y of x (cf. [2], Proposition 5.39 and Theorem 7.23). Geodesics are also helical in the Euclidean Received by the editors March 15, 1984 and, in revised form, June 20, 1984. 1980 Mathematics Subject Classification. Primary 53C40; Secondary 53C42.

Isometric immersions of Riemannian Spaces in Euclidean Spaces

Questions of the theory of isometric immersions of Riemannian spaces in Euclidean spaces beginning with the very first results onthis topic and also results on immersions of pseudo-Riemannian spaces in pseudo-Euclidean spaces and applications of the theory of immersions in the general theory of relativity are considered.

Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics I

We give a theory of oscillatory integrals in infinitely many dimensions which extends, for a class of phase functions, the finite dimensional theory. In particular we extend the method of stationary phase, the theory of Lagrange immersions and the corresponding asymptotic expansions to the infinite dimensional case. A particular application of the theory to the Feyman path integrals defined in previous work by the authors yields asymptotic expansions to all orders of quantum mechanical quantities in powers of Planck's constant.