Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R ⁿ . In these formulas, p-planes are represented as the column space of n×p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications – computing an invariant subspace of a matrix and the mean of subspaces – are worked out.     

On the axioms of planes in quaternionic geometry

Summary The axioms of planes in Riemannian geometry and Kaehlerian geometry have been largely studied. In this paper we study axioms for three kinds of planes in Quaternionic geometry: the axiom of quaternionic 4-planes, the axiom of half-quaternionic planes and the axiom of totally real planes. We also give a characterization of quaternionspaceforms in terms of the constancy of the totally real sectional curvatures.     

Real Hypersurfaces in Quaternionic Space Forms Satyisfying Axioms of Planes

A Riemannian manifold satisfies the axiom of 2-planes if at each point, there are suficiently many totally geodesic surfaces passing through that point. Real hypersurfaces in quaternionic space forms admit nice families of tangent planes, namely, totally real, half-quaternionic and quaternionic. Several definitions of axiom of planes arise naturally when we consider such families of tangent planes. We are able to classify real hypersurfaces in quaternionic space forms satisfying these definitions.     

Periodicity of residues of the number of finite labeled To-topologies

Let T₀(n) be the number of labeled topologies on points satisfying the T_o separation axiom. It is proved that for any prime p the sequence of residue classes T_o(n) mod p is periodic with period length p–1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 32–36, 1982.     

Newton's method for sections on Riemannian manifolds: Generalized covariant -theory

One kind of the L-average Lipschitz condition is introduced to covariant derivatives of sections on Riemannian manifolds. A convergence criterion of Newton's method and the radii of the uniqueness balls of the singular points for sections on Riemannian manifolds, which is independent of the curvatures, are established under the assumption that the covariant derivatives of the sections satisfy this kind of the L-average Lipschitz condition. Some applications to special cases including Kantorovich's condition and the γ-condition as well as Smale's α-theory are provided. In particular, the result due to Ferreira and Svaiter [Kantorovich's Theorem on Newton's method in Riemannian manifolds, J. Complexity 18 (2002) 304–329] is extended while the results due to Dedieu Priouret, Malajovich [Newton's method on Riemannian manifolds: covariant alpha theory, IMA J. Numer. Anal. 23 (2003) 395–419] are improved significantly. Moreover, the corresponding results due to Alvarez, Bolter, Munier [A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math. to appear] for vector fields and mappings on Riemannian manifolds are also extended.     

Calibrated Embeddings in the Special Lagrangian and Coassociative Cases

Every closed, oriented, real analytic Riemannian3–manifold can be isometrically embedded as a specialLagrangian submanifold of a Calabi–Yau 3–fold, even as thereal locus of an antiholomorphic, isometric involution. Every closed,oriented, real analytic Riemannian 4–manifold whose bundle of self-dual2–forms is trivial can be isometrically embedded as a coassociativesubmanifold in a G₂-manifold, even as the fixed locus of ananti-G₂ involution.These results, when coupledwith McLean's analysis of the moduli spaces of such calibratedsubmanifolds, yield a plentiful supply of examples of compact calibratedsubmanifolds with nontrivial deformation spaces.     

The theory of Zermelo-Fraenkel sets with Hilbert ɛ-terms

In this paper we study the role of functioning axioms on the deductive power of the system obtained from the Zermelo-Fraenkel ZF system by the introduction of -terms with the possibility of using them as a scheme for the substitution axiom. It is proved that if the system has a founding axiom the introduction of -terms does not extend the class of ZF theorems, while if the founding axiom is absent, there is an extension of the ZF theorems.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 569–575, November, 1972.     

Probleme der lokalen und globalen Mehrdimensionalen Differentialgeometrie

The following note deals with local and global problems of n-dimensional Riemannian manifolds immersed in Euclidean (n+m)-space, m1. The aim of the local part (1–6) is to discuss the relationship of a second and a third fundamental form to the intrinsic geometry; we derive interesting inequalities between some of their curvature invariants (3. 28); in 5–6 we consider special classes of submanifolds. The second part (7–11) deals with global problems: integral formulas of Minkowskian type, characterizations of submanifolds of spheres and submanifolds of constant mean curvature.     

Minimal submanifolds in a Riemannian manifold with pinched positive curvature

In this paper, we prove the following result: LetM be ann-dimensional compact curvature-invariant minimal subinanifoid immersed in a(> 2n–2/5n–2)- pinched Riemannian manifoldN. Denote by the second fundamental form ofM. If ¦¦²<3/9((5n–2)–2(n–1)), thenM is totally geodesic. This result generalizes the Simons pinching theorem.Supported by the JSPS postdoctoral fellowship and NSF of China.     

The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu ^{(n+2)/(n–2)} on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.     

On the measures of maximal entropy for the geodesic flow

Let M be a closed and connected manifold equipped with a C^∞ Riemannian metric. Using the geodesic arc formalism developed by Maé (J. Differential Geom. 45 (1997) 74–93) and Paternain [Ann. Sci. École Norm. Sup. 4eme série t 33 (2000) 121–138], we give a way of constructing a measure of maximal entropy for the geodesic flow.     

Geodesic mappings of 3-symmetric Riemannian spaces

It is proved that 3-symmetric Riemannian spaces of nonconstant curvature as well as Ricci 3-symmetric Riemannian spaces different from Einstein spaces do not admit nontrivial geodesic mappings.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 80–83, 1991.     

The vanishing rate of Weil–Petersson sectional curvatures

Geometriae Dedicata - A Riemannian manifold is called almost positively curved if the set of points for which all 2-planes have positive sectional curvature is open and dense. We find three new...     

An inversion formula for the Segal–Bargmann transform on a symmetric space of non-compact type

We prove analogs of the heat kernel transform inversion formulae of Golse, Leichtnam and the author [E. Leichtnam, F. Golse, M. Stenzel, Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds, Ann. Sci. École Norm. Sup. (4) 29 (6) (1996) 669–736. MR1422988 (97h:58153), Theorems 0.3, 0.4] in the setting of a Riemannian symmetric space of Helgason's non-compact type.     

Existence of solutions for variational inequalities on Riemannian manifolds

We establish the existence and uniqueness results for variational inequality problems on Riemannian manifolds and solve completely the open problem proposed in [S.Z. Németh, Variational inequalities on Hadamard manifolds, Nonlinear Anal. 52 (2003) 1491–1498]. Also the relationships between the constrained optimization problem and the variational inequality problems as well as the projections on Riemannian manifolds are studied.     

The Riemannian Structure of John and Uniform Domains in ℝ<Superscript>n</Superscript>

Using the theory of pre-ends, we study the boundary and metric properties of John and uniform domains in the Euclidean n-space. We obtain some results on the metric Riemannian structure of these classes of domains. We prove that the family of John domains is closed under the class of homeomorphisms quasi-isometric in the intrinsic Riemannian metric and the family of uniform domains is closed under the class of bi-Lipschitz mappings.Original Russian Text Copyright © 2005 Karmazin A. P.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 786–804, July–August, 2005.     

Foliations

The survey is based on works on the theory of foliations reviewed in RZhMatematika during 1970–1979. The basic topics are the classification of foliations, characteristic classes, the qualitative theory of foliations (holonomy, growth of leaves, etc.), and special classes of foliations (compact foliations, Riemannian foliations, etc.).Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 18, pp. 151–213, 1981.     

A metric on the manifold of immersions and its Riemannian curvature

E. Binz [1] considered two canonical Riemannian metrics on the space of embeddings of a closed (n–1) dimensional manifold into ⁿ , and computed the geodesic sprays. Here we consider the space of immersions Imm (M, N) whereM is without boundary, and we compute the covariant derivative (in the form of its connector) and the Riemannian curvature of one of these metrics, the non trivial one. The setting is close to that used byP. Michor [2], and we refer the reader to this paper for notation.     

On the visibility axiom for a two-dimensional Hadamard manifold

We consider a two-dimensional Hadamard manifoldM (i.e., a complete simply connected smooth Riemannian manifold of nonpositive Gaussian curvature). It is proved thatM satisfies the visibility axiom (for any>0 and each pointpM there exists a numberr=r(p, ) such that any geodesic whose distance fromp is at leastr subtends an angle less than or equal to from the pointp), if for any asymptotic geodesics and the inequality inf {d((t),(s))}=0 holds, whered is the Riemannian metric ofM.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 126–132.