Spaces of Riemannian metrics

In this paper, we consider spaces M of Riemannian metrics on a closed manifold M. In the case where the manifold M is equipped with a symplectic or contact structure, we consider spaces AM of associated metrics. We study geometric and topological properties of these spaces and Riemannian functionals on spaces of metrics. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 31, Geometry, 2005.     

On the van Kampen-Flores theorem

In this paper we generalize the van Kampen-Flores theorem for mappings of a simplex into a topological manifold.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 663–670, May, 1996.This research was supported by the International Science Foundation under grant No. JH2100.     

On the hyperbolicity criterion for noncompact Riemannian manifolds of special type

In this paper we study the behavior of bounded harmonic functions on complete Riemannian manifolds (of a certain special type) depending on the geometry of the manifold. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 557–563, April, 1996.     

On the geodesic curvature of an integral curve of a two-dimensional pfaffian manifold inE 4

The concept of generalized geodesic curvature is introduced along with the invariant mean geodesic curvature vector for a two-dimensional Pfaffian manifold inE ₄, and the indicatrix of generalized geodesic curvature of integral curves of this manifold is constructed.It is proved that the ratio of the fourth quadratic form of the manifold to its first fundamental form is equal to the square of the modulus of the generalized geodesic curvature vector, and an analog of a theorem of Euler is deduced.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 22–26.     

Boundary control and quasiphotons in the problem of reconstruction of a Riemannian manifold via dynamic data

The problem of reconstruction of a Riemannian manifold via a nonstationary response operator is considered. An approach to solving this problem which is based on methods of boundary control theory is suggested. The procedure of reconstruction makes use of nonstationary Gaussian beams (quasiphotons). Locality is a significant feature of the procedure. A class of manifolds which can be reconstructed within the framework of this approach is described in terms of boundary control theory. For example, this class includes all analytic manifolds. Bibliography: 21 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 21–50. Translated by A. P. Katchalov.     

The inverse problem of determining the source in the stationary transport equation on a riemannian manifold

The transport equation with an unknown right-hand side is considered on a compact Riemannian manifold. The right-hand side of this equation is recovered from values of the outcoming flow. Assumptions under which the solution of the inverse problem is unique are formulated.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 236–242.     

On the Holmgren-John uniqueness theorem for the wave equation with piecewise analytic coefficients

In this paper a uniqueness theorem is proved for the wave equation in the domain Q^2T=Ω×(0,2T), where Ω is a piecewise analytic Riemannian manifold (Riemannian polyhedron). Initial data are assumed to be given on a part Γ₀ × (0, 2T) of the space-time boundary of the cylinder Q^2T, Γ₀. The uniqueness of a weak solution is proved “in the large,” in a domain formed by the corresponding characteristics of the wave equation. Bibliography:24 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 113–136. Translated by T. N. Surkova.     

An invariant of triangulated links from the quantum dilogarithm

An invariant of triangulated links in triangulated three-dimensional manifolds is constructed by means of the cyclic quantum dilogarithm. Apparently, it is an isotopic invariant of nonoriented links in an oriented closed three-dimensional manifold. Bibliography: 13 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 208–214. Translated by B. Bekker.     

Intrinsic normalizations of a hyperplane distribution on the Grassmann manifold. I

We consider intrinsic normalizations of a hyperplane distribution on the Grassmann manifold of ann-dimensional projective space. Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 257–273, April–June, 1999. Translated by V. Mackevičius     

Equidistribution of divisors for sequences of holomorphic curves

We study holomorphic curves in ann-dimensional complex manifold on which a family of divisors parametrized by anm-dimensional compact complex manifold is given. If, for a given sequence of such curves, their areas (in the induced metric) monotonically tend to infinity, then for every divisor one can define adefect characterizing the deviation of the frequency at which this sequence intersects the divisor from the average frequency (over the set of all divisors). It turns out that, as well as in the classical multidimensional case, the set of divisors with positive defect is very rare. (We estimate how rare it is.) Moreover, the defect of almost all divisors belonging to a linear subsystem is equal to the mean value of the defect over the subsystem, and for all divisors in the subsystem (without any exception) the defect is not less than this mean value. This research was supported by RFBR grant No. 98-01-00867. Vladimir State Pedagogical University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 3, pp. 17–25, July–September, 2000. Translated by A. I. Shtern     

Isometric equivalence of directions in Riemannian symmetric spaces

A totally geodesic 2-surface with nonzero Gauss curvature in the real Grassmann manifold is considered. It is proved that any two one-dimensional tangent directions of the surface are put into coincidence by an isometry of the Grassmann manifold. Bibliography: 1 title. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 56–57.     

On the global theory of projective mappings

We consider the theory of constant rank projective mappings of compact Riemannian manifolds from the global point of view. We study projective immersions and submersions. As an example of the results, letf:(M, g) → (N, g′) be a projective submersion of anm-dimensional Riemannian manifold (M, g) onto an (m−1)-dimensional Riemannian manifold (N, g′). Then (M, g) is locally the Riemannian product of the sheets of two integrable distributions Kerf _* and (Kerf _*)^⊥ whenever (M, g) is one of the two following types: (a) a complete manifold with Ric ≥ 0; (b) a compact oriented manifold with Ric ≤ 0. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 111–118, July, 1995. This work was partially supported by the Russian Foundation for Basic Research grant No. 94-01-0195.     

A parabolic equation on a manifold of nonpositive curvature

For a fundamental solution of a parabolic equation on an n-dimensional Riemannian manifold we obtain two-sided estimates in terms of the curvature of the manifold.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 66–69.     

Geometry of distributions of flags on Grassmann manifolds of a projective space. I

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 214–227, April–June, 2000. Translated by V. Mackevičius     

Geometry of distributions of flags on Grassmann manifolds of a projective space. II

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 335–349, July–September, 2000. Translated by V. Mackevičius     

Wavelet decompositions on a manifold

A general method for constructing chains of embedded spline spaces on a smooth (not necessarily compact) manifold is suggested. A wavelet decomposition is obtained for the case of an arbitrary vector space. The results are illustrated by constructing the wavelet decompositon of a chain of embedded spaces of the B _φ-splines of zero order on a smooth manifold. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 346, 2007, pp. 26–38.     

Geometry of real grassmann manifolds. Parts I, II

The paper deals with the properties of the exterior algebra ℝ(Λⁿ) related to the Euclidean structure on ℝ(Λⁿ) induced by the scalar product in ℝ(Λⁿ). A geometric interpretation of inner multiplication for simple p-vectors is given. An invariant form of the Cartan criterion for the simplicity of a p-vector is given. The Plücker model realizing the real Grassmann manifold as a submanifold of the Euclidean space ℝ(Λⁿ), and an isometry of this submanifold onto the classical Grassmann manifold with SO(n)-invariant metric are described. A canonical decomposition of bivectors is given. Bibliography: 12 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 84–107. Translated by N. Yu: Netsvetaev.     

On the renormalized volumes for conformally compact Einstein manifolds

We study the renormalized volume of a conformally compact Einstein manifold. In even dimensions, we derive the analogue of the Chern-Gauss-Bonnet formula incorporating the renormalized volume. When the dimension is odd, we relate the renormalized volume to the conformal primitive of the Q-curvature. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 17, Differential and Functional Differential Equations. Part 3, 2006.     

Reconstruction of 3-submanifolds of large codimension in euclidean spaces from their gauss image

We present necessary and sufficient conditions for a regular three-dimensional manifold in the Grassmannian manifoldG(m, m+3) to be the Gauss image of a regular 3-submanifold in (m+3)-dimensional Euclidean space form>4. Translated fromMatemalicheskie Zametki, Vol. 62, No. 5, pp. 694–699, November, 1997. Translated by S. K. Lando     

Quantization of supermanifolds and an analytic proof of the Atiyah-Singer index theorem

An analytic proof of the Atiyah-Singer index, theorem is given with the help of the tools of supermathematics. The index formula for the Dirac operator on a spinor manifold is obtained here by direct calculation. A large portion of the paper is devoted to questions of quantization on supermanifolds, using spinors as example.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 38, pp. 3–118, 1991.