Geometry of the Normal Bundle of a Submanifold*

 We study the geometric behavior of the normal bundle T ^⊥ M of a submanifold M of a Riemannian manifold . We compute explicitely the second fundamental form of T ^⊥ M and look at the relation between the minimality of T ^⊥ M and M. Finally we show that the Maslov forms with respect to a suitable connection of the pair (T ^⊥ M, are null. Received March 14, 2001; in revised form February 11, 2002     

Maximality of analytic operator algebras

LetM be a σ-finite von Neumann algebra andα be an action ofR onM. LetH ^∞(α) be the associated analytic subalgebra; i.e.H ^∞(α)={X ∈M: sp_∞(X) [0, ∞]}. We prove that every σ-weakly closed subalgebra ofM that containsH ^∞(α) isH ^∞(γ) for some actionγ ofR onM. Also we show that (assumingZ(M)∩M ^α = Ci)H ^∞(α) is a maximal σ-weakly closed subalgebra ofM if and only if eitherH ^∞(α)={A ∈M: (I−F)xF=0} for some projectionF ∈M, or sp(α)=Γ(α).     

Weak helix submanifolds of euclidean spaces

Let M⊂ℝ ⁿ be a submanifold of a euclidean space. A vector d∈ℝ ⁿ is called a helix direction of M if the angle between d and any tangent space T _p M is constant. Let ℋ(M) be the set of helix directions of M. If the set ℋ(M) contains r linearly independent vectors we say that M is a weak r-helix. We say that M is a strong r-helix if ℋ(M) is a r-dimensional linear subspace of ℝ ⁿ . For curves and hypersurfaces both definitions agree. The object of this article is to show that these definitions are not equivalent. Namely, we construct (non strong) weak 2-helix surfaces of ℝ⁴. The author is supported by the Project M.I.U.R. “Riemann Metrics and Differentiable Manifolds” and by G.N.S.A.G.A. of I.N.d.A.M., Italy.     

Circularm-functions

Circularm-functions are introduced on smooth manifolds with boundary. We study the distribution of their critical circles and construct an example of a four-dimensional manifoldM ⁴ with boundary ∂M ⁴ that satisfies the condition ξ(∂M ⁴)=ξ(M ⁴,∂M ⁴)=0 but does not contain any circularm-function. We prove that a manifold with boundaryM ⁿ (n≥5) such that ξ(∂M ⁿ , ∂M ⁿ )=0 always contains a circularm-function without critical points in the interior manifold. Sukhumi Branch of the Tbilisi University, Sukhumi. Translated from Ukrainskii Matermaticheskii Zhurnal, Vol. 46, No. 6, pp. 776–781, June, 1994.     

Semi-primary hereditary rings

A semi-primary hereditary ring Σ, with radicalM and residue ring Γ=Σ/M, is uniquely determined by Γ and a Γ-bimoduleA=M/M ², whenever Σ admits a splitting Σ=Γ+A+M ².     

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.     

Central extensions of current groups

 In this paper we study central extensions of the identity component G of the Lie group C ^∞ (M,K) of smooth maps from a compact manifold M into a Lie group K which might be infinite-dimensional. We restrict our attention to Lie algebra cocycles of the form ω(ξ,η)=[κ(ξ,dη)], where κ:𝔨×𝔨→Y is a symmetric invariant bilinear map on the Lie algebra 𝔨 of K and the values of ω lie in Ω¹(M,Y)/dC ^∞ (M,Y). For such cocycles we show that a corresponding central Lie group extension exists if and only if this is the case for M=𝕊¹. If K is finite-dimensional semisimple, this implies the existence of a universal central Lie group extension of G. The groups Diff(M) and C ^∞ (M,K) act naturally on G by automorphisms. We also show that these smooth actions can be lifted to smooth actions on the central extension if it also is a central extension of the universal covering group G˜ of G. Received: 11 April 2002 / Revised version: 28 August 2002 / Published online: 28 March 2003     

On generalization of ⊕-cofinitely supplemented modules

We study the properties of ⊕-cofinitely radical supplemented modules, or, briefly, cgs ^⊕-modules. It is shown that a module with summand sum property (SSP) is cgs ^⊕ if and only if M/w Loc^⊕ M (w Loc^⊕ M is the sum of all w-local direct summands of a module M) does not contain any maximal submodule, that every cofinite direct summand of a UC-extending cgs ^⊕-module is cgs ^⊕, and that, for any ring R, every free R-module is cgs ^⊕ if and only if R is semiperfect.     

Lie jets and symmetries of prolongations of geometric objects

The Lie jet L _θ λ of a field of geometric objects λ on a smooth manifold M with respect to a field θ of Weil A-velocities is a generalization of the Lie derivative L _v λ of a field λ with respect to a vector field v. In this paper, Lie jets L _θ λ are applied to the study of A-smooth diffeomorphisms on a Weil bundle T ^A M of a smooth manifold M, which are symmetries of prolongations of geometric objects from M to T ^A M. It is shown that vanishing of a Lie jet L _θ λ is a necessary and sufficient condition for the prolongation λ ^A of a field of geometric objects λ to be invariant with respect to the transformation of the Weil bundle T ^A M induced by the field θ. The case of symmetries of prolongations of fields of geometric objects to the second-order tangent bundle T ² M are considered in more detail.     

A Remark on a Theorem by Kodama and Shimizu

We prove a characterization theorem for the unit polydisc Δ ⁿ ⊂ℂ ⁿ in the spirit of a recent result due to Kodama and Shimizu. We show that if M is a connected n-dimensional complex manifold such that (i) the group Aut (M) of holomorphic automorphisms of M acts on M with compact isotropy subgroups, and (ii) Aut (M) and Aut (Δ ⁿ ) are isomorphic as topological groups equipped with the compact-open topology, then M is holomorphically equivalent to Δ ⁿ .