Submanifolds Of Codimension Research Articles

The decomposition and classification of radiant affine 3-manifolds

An affine manifold is a manifold with torsion-free flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with holonomy consisting of affine transformations fixing a common fixed point. We decompose an orientable closed radiant affine 3-manifold into radiant 2-convex affine manifolds and radiant concave affine 3-manifolds along mutually disjoint totally geodesic tori or Klein bottles using the convex and concave decomposition of real projective $n$-manifolds developed earlier. Then we decompose a 2-convex radiant affine manifold into convex radiant affine manifolds and concave-cone affine manifolds. To do this, we will obtain certain nice geometric objects in the Kuiper completion of holonomy cover. The equivariance and local finiteness property of the collection of such objects will show that their union covers a compact submanifold of codimension zero, the complement of which is convex. Finally, using the results of Barbot, we will show that a closed radiant affine 3-manifold admits a total cross section, confirming a conjecture of Carri\`ere, and hence every radiant affine 3-manifold is homeomorphic to a Seifert fibered space with trivial Euler number, or a virtual bundle over a circle with fiber homeomorphic to a torus.

Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds

Introduction Singular integrals on Lipschitz submanifolds of codimension one Estimates on fundamental solutions General second-order strongly elliptic systems The Dirichlet problem for the Hodge Laplacian and related operators Natural boundary problems for the Hodge Laplacian in Lipschitz domains Layer potential operators on Lipschitz domains Rellich type estimates for differential forms Fredholm properties of boundary integral operators on regular spaces Weak extensions of boundary derivative operators Localization arguments and the end of the proof of Theorem 6.2 Harmonic fields on Lipschitz domains The proofs of the Theorems 5.1-5.5 The proofs of the auxiliary lemmas Applications to Maxwell's equations on Lipschitz domains Analysis on Lipschitz manifolds The connection between $d_\partial$ and $d_{\partial\Omega}$ Bibliography.

Conformal deformations of submanifolds in codimension two

Let Mn⊂Rn+2,n≥7, be a conformally deformable submanifold of euclidean space in codimension two. In this paper we show that if the submanifold has sufficiently low conformal nullity, a generic conformal condition, then it can be realized as a hypersurface of a conformally deformable hypersurface. The latter have been classified by Cartan early this century. Furthermore, it turns out that all deformations of the former are induced by deformations of the latter.

On Bojarski’s index formula for nonsmooth interfaces

Let D D be a Dirac type operator on a compact manifold M {\mathcal {M}} and let Σ \Sigma be a Lipschitz submanifold of codimension one partitioning M {\mathcal {M}} into two Lipschitz domains Ω ± \Omega _{\pm } . Also, let H ± p ( Σ , D ) {\mathcal {H}}^{p}_{\pm }(\Sigma ,D) be the traces on Σ \Sigma of the ( L p L^{p} -style) Hardy spaces associated with D D in Ω ± \Omega _{\pm } . Then ( H − p ( Σ , D ) , H + p ( Σ , D ) ) ({\mathcal {H}}^{p}_{-}(\Sigma ,D),{\mathcal {H}}^{p}_{+}(\Sigma ,D)) is a Fredholm pair of subspaces for L p ( Σ ) L^{p}(\Sigma ) (in Kato’s sense) whose index is the same as the index of the Dirac operator D D considered on the whole manifold M {\mathcal {M}} .

Euclidean conformally flat submanifolds in codimension two obtained as intersections

We characterize a large family of codimension two euclidean conformally flat submanifolds in the class of isometrically rigid ones and construct explicit examples. In [DF] we showed that generic conformally flat submanifolds with codimension two in euclidean space, f: M -* R+2, n > 5, can be divided into three classes, namely, the surface-like ones, those which admit locally a continuous 1-parameter family of isometric deformations, and those which are locally isometrically rigid. In addition, we generated explicit examples of elements in the first and second classes. On the other hand, no example came out from our rather elaborate descriptions of submanifolds in the third class, thus naturally raising the question of whether such submanifolds really exist. Our main achievement here is give a positive answer to this question. In what follows we make free use of definitions and results in [DF]. Our approach is the following. First we give a parametric description of a large family of generic conformally flat submanifolds, each of which is completely determined by two curves and two functions in one variable. Then, we prove that almost all elements in the family are locally isometrically rigid. Next, we give a brief indication of how they can be obtained by a geometric procedure, namely, as intersections starting from two flat hypersurfaces. We conclude this note by showing that a particular selection of curves and functions yields explicit examples. Consider a pair of smooth curves aj: Ij C R -, Rn+2, 1 1 everywhere. Now take V2 = I, x 12 small enough so that (1) 0(u,v) = 0(u,O) + jOv (us)ds, 0(u,O) = -I|a (u) 1, is positive, Ov being the unique (and necessarily smooth) solution of the integral equation of Volterra type Ov(u, ) =(u), a2(V)) 0(U, O) fa (2(v), a2(t))dt (2) J O j (j; (al1 (v), a2(t)) dt) Ov (u, s)ds. Received by the editors January 26, 1996 and, in revised form, April 30, 1997. 1991 Mathematics Subject Classification. Primary 53B25; Secondary 53C25.

Homogeneous Submanifolds of Codimension Two

We study codimension 2 homogeneous submanifolds of Euclidean space for which the index of minimum relative nullity is small. We prove that if minx∈Mνf(x)≤n-5, where ν(x) denotes the nullity of the second fundamental form of the immersion f at the point x, then the manifold M n is either isometric to a sphere or to a product of two spheres S2×S n−2 or covered by the Riemannian product S n−1 ×R. As a consequence, we obtain a classification of compact codimension 2 homogeneous submanifolds of dimension at least 5.

Some applications of a new integral formula for $∂̅_{b}$

Let M be a smooth q-concave CR submanifold of codimension k in $ℂ^n$. We solve locally the $∂̅_{b}$-equation on M for (0,r)-forms, 0 ≤ r ≤ q-1 or n-k-q+1 ≤ r ≤ n-k, with sharp interior estimates in Hölder spaces. We prove the optimal regularity of the $∂̅

Real Kaehler submanifolds in low codimension

Isometric immersions of Kaehler manifolds into euclidean space with real codimension 1 have been completely described, in codimension 2 at least in the complete minimal case. Codimension 3 examples are immediately obtained as complex hypersurfaces of Kaehler submanifolds in codimension one. The main purpose of this paper is to show that all local examples arise this way generically.

Groups of obstructions to surgery and splitting for a manifold pair

The surgery obstruction groups of manifold pairs are studied. An algebraic version of these groups for squares of antistructures of a special form equipped with decorations is considered. The squares of antistructures in question are natural generalizations of squares of fundamental groups that occur in the splitting problem for a one-sided submanifold of codimension 1 in the case when the fundamental group of the submanifold is mapped epimorphically onto the fundamental group of the manifold. New connections between the groups , the Novikov-Wall groups, and the splitting obstruction groups are established.

On deformations product

The produet of deformations in Riemannian manifolds is defined. The produet of infînitesimal isometric deformations (IID) is considered and some results of submanifolds of codimension > 2 are established. We also study the second fundamental form on the produet manifold and prove that the produet X of two hypersurfaces

Collision–Ejection Manifold and Collective Analytic Continuation of Simultaneous Binary Collisions in the PlanarN-Body Problem

We study simultaneous binary collision (SBC) singularities ofLbinaries in the planarN-body problem, 2L≤N. We introduce the generalized Levi-Civita transformation and follow it by a new transformation which we call the projective transformation near a SBC singularity. We use this transformation to show the following near a SBC singularity:(1) In the generalized Levi-Civita variables, near a SBC singularity, the collection of collision and ejection orbits together with the singularity form a real analytic submanifold, which we call the collision–ejection (CE) manifold.(2) LetRc(Re) be the collection of SBC (SBE) orbits in phase space, i.e., in the original variables. Then, bothRcandReare real analytic.(3) Each collision orbit corresponds to a unique ejection orbit. Together, they form a real analytic orbit in the generalized Levi-Civita variables, which we call a collision–ejection orbit.(4) LetC⊂Rc(E⊂Re) be a submanifold of initial conditions that end (start) in a simultaneous binary collision (ejection) singularity. We show thatC(E) can be chosen to be a real analytic submanifold of codimension 1 inRc(Re), and that the correspondence in item (3) above defines a real analytic section mapping fromCtoE.(5) Collision and ejection orbits can be collectively analytically continued, i.e., each collision–ejection orbit can be written as a convergent power series int1/3, with coefficients that depend real analytically on the initial conditions inC.(6) SBC orbits ofJ<Lbinaries do not accumulate on any SBC orbit ofLbinaries.(7) A single binary collision singularity in theN-body problem is real analytic block-regularizable.We also give the asymptotic behaviour of collision and ejection orbits.

On Ricci Curvatures of Hypersurfaces in Abstract Wiener Spaces

Several concrete examples of hypersurfaces, i.e., submanifolds of codimension 1, in abstract Wiener space will be given to study how the signs of their Ricci curvatures varies. As a result, it will be concluded that the Ricci curvature is no longer a geometrical object in contrast with the curvatures of finite dimensional manifolds.

The Weierstrass representation for complete minimal real Kaehler submanifolds of codimension two

The Weierstrass representation for complete minimal real Kaehler submanifolds of codimension two

Isometric deformations of compact Euclidean submanifolds in codimension 2

Isometric deformations of compact Euclidean submanifolds in codimension 2

Symplectomorphic codimension 1 totally geodesic submanifolds

We show that the coisotropic totally geodesic properly embedded submanifolds of codimension 1 of a simply connected complete Kähler manifold of non-positive sectional curvature are symplectically linearizable.

Mayer-Vietoris formula for the determinant of a Laplace operator on an even-dimensional manifold

Let Δ \Delta be a Laplace operator acting on differential p-forms on an even-dimensional manifold M. Let Γ \Gamma be a submanifold of codimension 1. We show that if B is a Dirichlet boundary condition and R is a Dirichlet-Neumann operator on Γ \Gamma , then Det ( Δ + λ ) = Det ( Δ + λ , B ) Det ( R + λ ) {\operatorname {Det}}(\Delta + \lambda ) = {\operatorname {Det}}(\Delta + \lambda ,B){\operatorname {Det}}(R + \lambda ) and Det ∗ Δ = 1 ( det A ) 2 Det ( Δ , B ) Det ∗ R {\operatorname {Det}^ \ast }\Delta = \frac {1}{{{{(\det A)}^2}}}{\operatorname {Det}}(\Delta ,B){\operatorname {Det}^ \ast }R . This result was established in 1992 by Burghelea, Friedlander, and Rappeler for a 2-dimensional manifold with p = 0 p = 0 .

Enroulements asymptotiques du mouvement brownien autour de lacets dans une variété riemannienne compacte de dimension 3

Let M be a Riemannian compact manifold of dimension 3, X its Brownian motion, M̄ a compact submanifold of codimension 2, and ω a closed 1-form on M′ = M‘M̄; then the Stratonovitch integral t −1 ∫ 0 t ω( X s) converges in law towards a Cauchy variable.

Pointed helical submanifolds of euclidean space

A pointed helical submanifold of Euclidean space is defined and studied. In particular, complete pointed helical submanifolds of codimension 2 in a Euclidean space are classified.

Holomorphic automorphisms of quadrics

a R-valued Hermitian form. The following set is called a quadric Q = {(z, w) ∈ C : v = 〈z, z〉}. Q is presumed to be nondegenerate, i.e., i.) 〈z, b〉 = 0 for all z implies b = 0 ii.) The forms 〈·, ·〉 are linearly independent for j = 1, . . . , k. Any quadric Q is a homogeneous manifold (the transformations z∗ = p + z, w∗ = q + w + 2i〈z, p〉 with (p, q) ∈ Q act transitively on Q). Therefore, we restrict ourselves to the local automorphisms preserving a fixed point, say the origin. The subgroup of these automorphisms, being called isotropy group, is a finite dimensional Lie group iff Q is nondegenerate (see [2, 5]). The connected component of the unit of the isotropy group will be denoted by Aut0Q. Our goal is to find the explicit description for Aut0Q. This is important for the study of real submanifolds of codimension k in C. It follows from the results of Henkin, Tumanov and Forstneric [6, 11, 5] that any local CR-diffeomorphism of a nondegenerate quadric extends to a birational map of C with the degree uniformly bounded within the same (n, k). This result might be considered as a generalization of the Poincare-Alexander theorem [7, 1] about the extension of a local CR-diffeomorphism of a hyperquadric in C. However, explicit descriptions of the isotropy groups are only known for hyperquadrics (k = 1) and direct products of hyperquadrics.

On the definition of quantum free particle on curved manifolds

A self-consistent definition of quantum free particle on a generic curved manifold naturally emerges by restricting the dynamics to submanifolds of co-dimension one.