Banach manifolds of algebraic elements in the algebra $$\mathcal{L}$$ ( H) of bounded linear operatorsof bounded linear operators

Given a complex Hilbert space H, we study the manifold of algebraic elements in . We represent as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C^*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C^*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine connection ∇ are defined on M, and the geodesics are computed. If M is the orbit of a finite rank projection, then a G-invariant Riemann structure is defined with respect to which ∇ is the Levi-Civita connection. Supported by Ministerio de Educación y Cultura of Spain, Research Project BFM2002-01529.     

Bundle-connection pairs and loop group representations

LetM be a connected differentiable manifold. Denote by Ω _m (M) the space ofH ¹-loops based at a fixed pointm∈M. Associated to Ω _m (M) one has , the group of unparameterized loops. Given a bundle-connection pair (E,∇) overM with fiber the finite-dimensional vector spaceV and structure groupG⊂GL(V) we get (up to equivalence) a smooth representation of inG given by the parallel transport operatorP ^∇. It is possible to find in the literature several versions of the converse theorem, namely: all (smooth) representations of arise in the above described way from a bundle-connection pair. It is shown in the present paper that the correct setting for this theorem is the theory of induced representations for groupoids. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 503–518, April, 1997.     

Varietes symplectiques affines

We will be concerned with two open questions in symplectic geometry. The first question is the existence problem for lagrangian foliations. The second question is to know whether an affine manifold can be embedded as leaf of a lagrangian foliation. We prove that every homogeneous symplectic manifold with closed regular action of a solvable Lie group has lagrangian foliations. Moreover such manifold (M,) has a bilagrangian linear connection (M,) such that is parallel with respect to . About the second question, we prove that every connected and simply connected Lie group with left invariant affine structure can be embedded as leaf of a left invariant lagrangian foliation in a symplectic Lie group (G,).     

Flat Affine Surfaces in \mathbb{R}^4 with Flat Normal Connection

In this paper we study nondegenerate affine surfaces in the 4-dimensional affine space . We assume that both the connection and the normal connection induced by the canonical equiaffine transversal bundle are flat. Surfaces with constant equiaffine transversal bundle are trivial examples of such surfaces. Here, we obtain a complete classification of all such surfaces which do not have constant equiaffine normal bundle.     

Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature

Let (M, g) be an n-dimensional compact and connected Riemannian manifold of constant scalar curvature. If the sectional curvatures of M are bounded below by a constant α > 0, and the Ricci curvature satisfies Ric < (n − 1)αδ, δ ≥ 1, then it is shown that either M is isometric to the n-sphere Sⁿ(α) or else each nonzero eigenvalue λ of the Laplacian acting on the smooth functions of M satisfies the following:

Analytic functionals annihilated by ideals

Let V be a n-dimensional Stein manifold, I be a closed ideal of holomorphic functions on V. It was proved by Roger Gay that, given an analytic functional T such that hT=0 (as a functional) for any h∈I, one can find some (n, n)_compactly supported current , such that for any ϕ ∈Iɛ ^0,0(V) and T(h)= for any h analytic on V. In this paper, we give some explicit construction of Ť in terms of residual currents when I is defined as a complete intersection or is locally Cohen-Macaulay. Moreover, by means of integral representation formulas of the Andersson-Berndtsson-Passare type, we also study the non complete intersection case in order to represent analytic functionals orthogonal to the ideal in terms of currents annihilated (as currents) by some power (less than n) of the local integral closure ofIɛ^0,0.     

A finiteness theorem for transitive foliations and flat vector bundles

In this paper we prove a finiteness theorem for the spectral sequence (E _i(∇), (d _∇)_i) associated to a transitive foliation on a compact manifoldM, and to a flat vector bundleE overM with flat connection ∇. we also compute some examples of homogeneous Lie foliations on compact connected homogeneous spaces.     

Sharp Sobolev inequalities of second order

Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and ₂ ² (M) be the Sobolev space consisting of functions in L²(M) whose derivatives up to the order two are also in L²(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H ₂ ² (M),

Boundary value problems for nonuniformly elliptic equations with measurable coefficients

Summary Let A be a symmetric N × N real-matrix-valued function on a connected region in Rⁿ, with A positive definite a.e. and A, A⁻¹ locally integrable. Let b and c be locally integrable, non-negative, real-valued functions on Ω, with c positive a.e. Put a(u, v) = = ((A∇u, ∇v) + buv) dx. We consider in X the weak boundary value problem a(u, v) = = fvcdx, all v ε X; where X is a suitable Hilbert space contained in H _loc ^1,1 (Ω). Criteria are given in order that the Green's operator for this problem have an integral representation and bounded eigenfunctions; in addition, criteria for compactness are given. Entrata in Redazione il 21 giugno 1975. Research was partially supported by the National Science Foundation under Grant GP-28377A2.     

On totally geodesic affine immersions

In this paper, totally geodesic affine immersionsf: (M, ) are studied in the case when is an affine manifold of recurrent curvature. It is proved that(M, ) if flat or of recurrent curvature. And iff is additionally umbilical with the shape tensorA 0 and dimM 3, then(M, ) is locally projectively flat. Examples of such immersions are also stated.     

Combinatorial fiber bundles and fragmentation of a fiberwise PL homeomorphism

With a compact PL manifold X we associate a category . The objects of are all combinatorial manifolds of type X, and morphisms are combinatorial assemblies. We prove that the homotopy equivalence

Four-Dimensional Simply Connected Symplectic Symmetric Spaces

A symplectic is a symmetric space endowed with a symplectic structure which is invariant by the symmetries. We give here a classification of four-dimensional symplectic which are simply connected. This classification reveals a remarkable class of affine symmetric spaces with a non-Abelian solvable transvection group. The underlying manifold M of each element (M, ) belonging to this class is diffeomorphic to Rⁿwith the property that every tensor field on M invariant by the transvection group is constant; in particular, is not a metric connection. This classification also provides examples of nonflat affine symmetric connections on Rⁿwhich are invariant under the translations. By considering quotient spaces, one finds examples of locally affine symmetric tori which are not globally symmetric.     

The spectrum of compact hypersurface in sphere

Let M be a compact minimal hypersurface of sphere Sn 1(1). Let (M) be H (r)-torus of sphere Sn 1 (1).Assume they have the same constant mean curvature H, the result in [1] is that ifSpec0(M, g) =Spec0((M), g),then for 3≤ n ≤ 6, r2≤n-1/n or n ≥ 6, r2 ≥ n-1, then M is isometric to (M). We improved the result and prove that: if Spec0(M,g) =Spec0((M),g), then M is isometric to (M). Generally, if Specp(M,g) =Specp((M),g), here p is fixed and satisfies that n(n - 1) ≠ 6p(n - p), then M is isometric to (M).     

A new class of contact riemannian manifolds

N. Tanaka ([10]) defined the canonical affine connection on a nondegenerate integrable CR manifold. In the present paper, we introduce a new class of contact Riemannian manifolds satisfying (C) ( for any unit -geodesic ( , where is the generalized Tanaka connection. In particular, when the associated CR structure of a given contact Riemannian manifold is integrable we have a structure theorem and find examples which are neither Sasakian nor locally symmetric but satisfy the condition (C). This work was supported in part by BSRI 98-1425.     

Local rigidity for certain groups of toral automorphisms

Let Γ = SL(n, ℤ) or any subgroup of finite index, n ≥ 4. We show that the standard action of Γ on ⁿ is locally rigid, i.e., every action of Γ on ⁿ by C^∞ diffeomorphisms which is sufficiently close to the standard action is conjugate to the standard action by a C^∞ diffeomorphism. In the course of the proof, we obtain a global rigidity result (Theorem 4.12) for actions of free abelian subgroups of maximal rank in SL(n, ℤ). Partially supported by NSF grant DMS9011749.     

Abel-Lidskii bases in the non-self-adjoint inverse boundary problem

Let M be a manifold with boundary δM≠Ф. Let A be a second-order elliptic partial differential operator given on M. Denote by R_λ(x, y), x, y∈M, λε \σ(A) the Schwartz kernel of (A−λI)⁻¹. Consider the Gel'fand inverse boundary problem of the reconstruction of (M, A) via a given R_λ(x, y), x, y∈δM,λε . We prove that if the principal symbol of A satisfies some geometric condition (the Bardos-Lebeau-Rauch condition), then these data determine M uniquely, and they determine A to within the group of generalized gauge transformation on M. The above-mentioned geometric condition means, roughly speaking, that any geodesic (in the metric generated by A) leaves M. Bibliography: 29 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 161–190. Translated by Ya. Kurylev.     

Problem with Critical Sobolev Exponent and with Weight

The authors consider the problem: -div(p▽u) = uq-1 λu, u > 0 inΩ, u = 0 on (?)Ω, whereΩis a bounded domain in Rn, n≥3, p :Ω→R is a given positive weight such that p∈H1 (Ω)∩C(Ω),λis a real constant and q = 2n/n-2, and study the effect of the behavior of p near its minima and the impact of the geometry of domain on the existence of solutions for the above problem.     

A note on curvature of α-connections of a statistical manifold

The family of α-connections ∇^(α) on a statistical manifold equipped with a pair of conjugate connections and is given as . Here, we develop an expression of curvature R ^(α) for ∇^(α) in relation to those for . Immediately evident from it is that ∇^(α) is equiaffine for any when are dually flat, as previously observed in Takeuchi and Amari (IEEE Transactions on Information Theory 51:1011–1023, 2005). Other related formulae are also developed. The work was conducted when the author was on sabbatical leave as a visiting research scientist at the Mathematical Neuroscience Unit, RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan.     

Almost Extrinsically Homogeneous Submanifolds of Euclidean Space

Consider a closed manifold M immersed in R^m. Suppose that the trivial bundle M × R^m = T M ⊗ ν M is equipped with an almost metric connection ~ ∇ which almost preserves the decomposition of M × R^m into the tangent and the normal bundle. Assume moreover that the difference Γ = ∂~∇ with the usual derivative ∂ in R^m is almost ~∇-parallel. Then M admits an extrinsically homogeneous immersion into R^m. Mathematics Subject Classifications (2000): 53C20, 53C24, 53C30, 53C42, 53C40.     

Remarks on the Extremal Functions for the Moser-Trudinger Inequality

We will show in this paper that if A is very close to 1, then I（M,λ,m） =supu∈H0^1,n（m）,∫m｜△↓u｜^ndV=1∫Ω（e^αn｜u｜^n/（n-1）-λm∑k=1｜αnun/（n-1）｜k/k!）dV can be attained, where M is a compact-manifold with boundary. This result gives a counter-example to the conjecture of de Figueiredo and Ruf in their paper titled ＂On an inequality by Trudinger and Moser and related elliptic equations＂ （Comm. Pure. Appl. Math., 55, 135-152, 2002）.