Smooth N-dimensional Manifold Research Articles

О скалярных компонентах канонической формы на расслоениях реперов высших порядков

A detailed obtaining of the expressions for the scalar components of the canonical form on higher order frame bundles over a smooth manifold has been done. The canonical form on the frame bundle of order p + 1 over an n-dimensional smooth manifold is a vector-valued differential 1-form with values in the tangent space to the p-th order frame bundle over the n-di­mensional arithmetical space at the unit of the p-th order differential group. The scalar components of the canonical form are its coefficients with respect to natural basis of the tangent space. For every frame, there exists a polynomial mapping representing the frame in a given local chart on the manifold. Therefore, for any tangent vector to the frame bundle there is a first order Taylor expansion of one-parametric family of poly­nomial mappings representing the tangent vector. We obtain the formulas of the scalar components from the equations for coefficients of the two expansions for some tangent vector.

О конструкции канонической формы на расслоении реперов

The detailed description of the construction of the canonical form on the higher order frame bundle over an n-dimensional smooth manifold is given. In particular, it is shown that some vector space isomorphism play­ing the key role in this construction is defined correctly, i. e. it depends only on the frame of order p + 1 and does not depend on the choice of its representative, i. e. a local diffeomorphism which (p + 1)-jet is exactly this frame. This isomorphism acts from the direct sum of n-dimensional arithmetic space and the Lie algebra of the p-th order differential group to the tangent space to the p-th order frame bundle over the manifold at the p-th order frame lying “below”. The action of this isomorphism can be splitted into two its restrictions. The first one acts from the first direct sum­mand, and the second one acts from the second direct summand. It is shown that the first restriction depends only on the choice of the (p + 1)-fra­me, while the second one is closely related to fundamental vector fields and therefore does not depend of this frame at all.

Introducing totally nonparallel immersions

An immersion of a smooth n-dimensional manifold M→Rq is called totally nonparallel if, for every distinct x,y∈M, the tangent spaces at f(x) and f(y) contain no parallel lines; equivalently, they span a 2n-dimensional space. Given a manifold M, we seek the minimum dimension TN(M) such that there exists a totally nonparallel immersion M→RTN(M). In analogy with the totally skew embeddings studied by Ghomi and Tabachnikov, we find that totally nonparallel immersions are related to the generalized vector field problem, the immersion and embedding problems for real projective spaces, and nonsingular symmetric bilinear maps.Our study of totally nonparallel immersions follows a recent trend of studying conditions which manifest on the configuration spaceFk(M) of k-tuples of distinct points of M; for example, k-regular embeddings, k-skew embeddings, k-neighborly embeddings, and several others. Typically, a map satisfying one of these configuration space conditions induces some Sk-equivariant map on the configuration space Fk(M) (or on a bundle thereof) and obstructions can be computed in the form of Stiefel-Whitney classes. However, the existence problem for such conditions is relatively unstudied.Our main result is a Whitney-type theorem: every smooth n-manifold M admits a totally nonparallel immersion into R4n−1, one dimension less than given by genericity. We begin by studying the local problem, which requires a thorough understanding of the space of nonsingular symmetric bilinear maps, after which the main theorem is established using the removal-of-singularities h-principle technique due to Gromov and Eliashberg. When combined with a recent non-immersion theorem of Davis, we obtain the exact value TN(RPn)=4n−1 when n is a power of 2. This is the first optimal-dimension result for any closed manifold M besides S1, for any of the recently-studied configuration space conditions.

О дифференциальных уравнениях тензоров кривизны фундаментально-групповой и аффинной связностей

The principal bundle is considered, the base of which is an n-dimensional smooth manifold, and the typical fiber is an r-fold Lie group. Structure equations for the forms of the fundamental group and affine connections are given, each of which contains the corresponding components of the curvature tensor. For each connection, an approach is shown that allows to find the differential equations for the components of the curvature tensor of the corresponding connection in a faster way than by differentiating the expressions of these objects in terms of the connection objects and their Pfaffian derivatives. The method consists in successively solving cubic equations, first by Laptev’s lemma, then by Cartan’s lemma. Taking into account the comparisons modulo basic forms, we obtain already known results (see [3]). Thus, differential equations are derived for the components of the curvature tensor of the first-order fundamentalgroup connection, as well as for the components of the curvature tensor of the affine connection.

Subsequent singularities of mean convex mean curvature flows in smooth manifolds

For any n-dimensional smooth manifold \(\Sigma \), we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in \(\Sigma \) are cylindrical (of convex type) if the flow converges to a smooth hypersurface \(M_\infty \) (maybe empty) at infinity. Previously this was shown (1) for \(n\le 7\), and (2) for arbitrary n up to the first singular time without the smooth condition on \(M_\infty \).

A Geometric Problem and the Hopf Lemma. II

A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in ℝn+1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn+1 =constant in case M satisfies: for any two points (X′,Xn+1), $$ {\left( {{X}\ifmmode{'}\else$'$\fi,\ifmmode\expandafter\hat\else\expandafter\^\fi{X}_{{n + 1}} } \right)} $$ on M, with $$ X_{{n + 1}} > \ifmmode\expandafter\hat\else\expandafter\^\fi{X}_{{n + 1}} $$ , the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems.

Maximal degree variational principles and Liouville dynamics

Let M be smooth n-dimensional manifold, fibered over a k-dimensional submanifold B as π :M→B , and ϑ∈ Λ k ( M); one can consider the functional on sections φ of the bundle π defined by ∫ Dφ ∗(ϑ) , with D a domain in B. We show that for k= n−2 the variational principle based on this functional identifies a unique (up to multiplication by a smooth function) nontrivial vector field in M, i.e., a system of ODEs. Conversely, any vector field X on M satisfying X⌟ dϑ=0 for some ϑ∈ Λ n−2 ( M) admits such a variational characterization. We consider the general case, and also the particular case M= P× R where one of the variables (the time) has a distinguished role; in this case our results imply that any Liouville (volume-preserving) vector field on the phase space P admits a variational principle of the kind considered here.

Le rayon spectral essentiel de l'opérateur de Ruelle sur des variétés höldériennes

We study the deterministic and random Ruelle transfer operator L induced by an expanding map ƒ of a smooth n-dimensional manifold X and a bundle automorphism ϑ of a rank L vector bundle E. We prove the following exact formula for the essential spectral radius of L on the space C r, α of r-times continuously differentiable sections of E with α-H o ̈ lder r-th derivative: r ess( L;C r,α) = exp ( vϵrg sup {h v + λ v, − (r+α)x v}) , where Erg denotes the set of ƒ-ergodic measures, h v the entropy of ƒ with respect to v, λ v the largest Lyapunov exponent of the cocycle ϑ (x) = ϑ(ƒ k−1(x) ·…· ϑ(x) , and x v the smallest Lyapunov exponent of the differential Dƒ k(x) , x ϵ X, k = 1, 2,…. A similar result holds for the random case.

Thom polynomials for open Whitney umbrellas of isotropic mappings

A smooth mapping $f:L^n → (M^{2n},ω)$ of a smooth n-dimensional manifold L into a smooth 2n-dimensional symplectic manifold (M,ω) is called isotropic if f*ω vanishes. In the last ten years, the local theory of singularities of isotropic mappings has been

Degree formulas for maps with nonintegrable Jacobian

This paper arose from a discussion sparked between the authors after the lecture of Louis Nirenberg at the Conference in Naples on June 1, 1995. He presented a joint work with Haim Brezis [BN] on the degree theory for VMO (vanishing mean oscillation) mappings f : X → Y between n-dimensional smooth manifolds. Their results include a variety of discontinuous maps. We soon realized that we can contribute to their work by studying some Orlicz– Sobolev classes weaker than W (X,Y ). Our approach relies on new estimates for the Jacobians [IS], [GIM] and most recent improvements [I] concerning nonlinear commutators. Also L-Hodge theory [S], [ISS] plays a crucial role in this paper. Let us begin with the well known formula for the degree of a C-map f : X → Y :

Indeterminacy in general equilibrium economies with incomplete financial markets: Mixed asset returns

We consider the structure of the set of equilibrium allocations in a two-period, pure exchange economy with incomplete financial markets. There are N second period spots and 0< M< N assets. Asset returns are affine functions of spot commodity prices. Economies are parameterized by endowments and asset returns. We show that, for an open, dense set of economies, the set of equilibrium allocations contains a smooth N-dimensional manifold, provided that there are ‘enough’ consumers and that ( N− M) M≧ N.

Linear independence of Morse functions on manifolds

It is proved that on a smooth n-dimensional manifold there exist n linearly independent Morse functions.

Stabilization of Hamiltonian systems

1. HAMILTONIAN SYSTEMS IN THIS paper we will be concerned with the stabilization by feedback of Hamiltonian systems. In order to facilitate our discussions (especially when applying Lyapunov’s second method) we will restrict ourselves to a particular, although natural, subclass of Hamiltonian systems given in the following way [l]. Let Q be an n-dimensional smooth manifold, denoting the configuration space, and let T*Q be the cotangent bundle, denoting the phase space or srufe space. Furthermore there is a smooth m-dimensional output manifold Y (m < n) and a smooth output map C : Q- Y. (Smooth will mean C” or Ck, with k sufficiently big, although we shall restrict ourselves in the second part of Section 2 to analytic data.) For simplicity we take C to be submersive. so rank dC(q) = m. We assume that the system on T*Q has an internal energy which is the sum of a kinetic energy K and a porentiul energy V. This means that there exists a Riemannian metric (,) on Q, in local coordinates (q,, . . . , q,J for Q given by

Periodic solutions of Lagrangian systems on a compact manifold

Let M be a smooth n-dimensional manifold and let TM be its tangent bundle. We consider a time periodic Lagrangian of period T, l t : TM → R, and we seek T-periodic solutions of the Lagrange equations, which in local coordinates are d dt ∂L ∂ q ̇ (t,q, q ̇ )− ∂L ∂q (t,q, q ̇ )=0, i=1…n. (∗) Our main result states that if the fundamental group of M is finite, then (∗) has infinitely many T-periodic solutions, provided that l t satisfies certain physically reasonable assumptions.

Codimension one immersions and the Kervaire invariant one problem

Let i: M↬ℝn+1 be a self-transverse immersion of a compact closed smooth n-dimensional manifold in (n + 1)-dimensional Euclidean space. A point of ℝn+1 is an r-fold intersection point of the immersion if it is the image under i of (at least) r distinct points of the manifold. The self-transversality of i implies that the set of r-fold intersection points is the image of an immersion of a manifold of dimension n+1-r (the empty set if r &gt; n + 1). In particular, the set of (n + l)-fold intersection points is finite of order, say, θ(i). In this paper we are concerned with the set of values of θ(i) for (self-transverse) immersions of all (compact closed smooth) manifolds of given dimension n.

Imbeddings, Immersions, and Characteristic Classes of Differentiable Manifolds

Let $I_n^i$ be the set of $\bmod {\text { - }}2$ characteristic classes which are of dimension i, and they are zero for all n-dimensional smooth manifolds. Let $I_{n,k}^i$ be the set of i-dimensional $\bmod {\text { - }}2$ characteristic classes which are zero for all n-dimensional smooth manifolds which immerse in codimension k, (we are talking about normal characteristic classes). Let K be the (graded) ideal in ${H^ \ast }(BO,{Z_2})$ generated by ${w_{k + 1}},{w_{k + 2}}, \ldots$. Then if $i \leqslant (n + k)/2$, we have $I_{n,k}^i = I_n^i + {K^i}$. We have some related results for imbedded manifolds, and also for manifolds which immerse or imbed with an SO, U, SU, Spin, etc. structure on the normal bundle.

Imbeddings, immersions, and characteristic classes of differentiable manifolds

Let I n i I_n^i be the set of mod - 2 \bmod {\text { - }}2 characteristic classes which are of dimension i, and they are zero for all n-dimensional smooth manifolds. Let I n , k i I_{n,k}^i be the set of i-dimensional mod - 2 \bmod {\text { - }}2 characteristic classes which are zero for all n-dimensional smooth manifolds which immerse in codimension k, (we are talking about normal characteristic classes). Let K be the (graded) ideal in H ∗ ( B O , Z 2 ) {H^ \ast }(BO,{Z_2}) generated by w k + 1 , w k + 2 , … {w_{k + 1}},{w_{k + 2}}, \ldots . Then if i ⩽ ( n + k ) / 2 i \leqslant (n + k)/2 , we have I n , k i = I n i + K i I_{n,k}^i = I_n^i + {K^i} . We have some related results for imbedded manifolds, and also for manifolds which immerse or imbed with an SO, U, SU, Spin, etc. structure on the normal bundle.

An example of a local flow on a manifold

Let p be a point of a smooth n-dimensional manifold. If n is even it is easy to construct a local flow about p such that p is an isolated critical point and no orbit except the stationary one at p has p as a limit point. We call such a flow a nonnull flow about p (NN-flow). Mendelson has conjectured that NN-flows do not exist on odd dimensional manifolds. We show that Mendelson’s conjecture is false by constructing an NN-flow on any smooth manifold whose dimension is an odd integer exceeding one.

Integrably Parallelizable Manifolds

A smooth manifold ${M^n}$ is called integrably parallelizable if there exists an atlas for the smooth structure on ${M^n}$ such that all differentials in overlap between charts are equal to the identity map of the model for ${M^n}$. We show that the class of connected, integrably parallelizable, n-dimensional smooth manifolds consists precisely of the open parallelizable manifolds and manifolds diffeomorphic to the n-torus.

Integrably parallelizable manifolds

A smooth manifold M n {M^n} is called integrably parallelizable if there exists an atlas for the smooth structure on M n {M^n} such that all differentials in overlap between charts are equal to the identity map of the model for M n {M^n} . We show that the class of connected, integrably parallelizable, n-dimensional smooth manifolds consists precisely of the open parallelizable manifolds and manifolds diffeomorphic to the n-torus.