Study of pseudometric properties on domains in Nagano spaces.
problem Characterize pseudometrics on domains in real-type Nagano spaces.
method Analyze Kobayashi-type pseudometrics on domains, proving properties and computing specific cases.
result The pseudometric is a genuine metric under certain conditions and has specific properties in higher rank.
In this article, I classify the totally geodesic submanifolds in the complex 2-Grassmannians and in the quaternionic 2-Grassmannians. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank 2, published by Chen and Nagano (B.…
Introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional.
problem Deriving critical metrics for the Einstein-Hilbert functional on compact Riemannian manifolds.
method Restricts the variational problem to an infinite-dimensional subspace.
result Derives a novel structural characterization of critical metrics.
Extends topological results for nonpositive curvature spaces.
problem Characterize and understand topological properties of BNPC spaces.
method Extends results from CAT(0) spaces to BNPC spaces, using links and geodesic bicombings.
result Locally BNPC topological manifolds have discrete singular sets and are homeomorphic to Euclidean space.
New gauge preserves Einstein metrics' interactions, proving rigidity on negatively curved manifolds.
problem Stability and deformation theory of Einstein metrics.
method Introduces Chen-Nagano gauge condition, linking Lichnerowicz Laplacian to shifted scalar operator.
result Chen-Nagano gauge collapses to classical transverse-traceless gauge under spectral pinching assumptions.
By attaching a Lie algebra of germs of analytic vector fields to every point of a (real or complex) analytic variety V we construct the Nagano foliation of the variety. We prove that the Nagano foliation of V is a stratification. The treatment of the subject is totally coordinate free but relies on the Oka-Cartan-Serre…
Proves conjectures about maximal antipodal sets in symmetric and generalised symmetric spaces.
problem Cohomological descriptions of maximal antipodal sets in symmetric spaces.
method Equivariant cohomology theory.
result Proves several long-standing conjectures by Chen--Nagano and extends them to generalised symmetric spaces.
The paper defines generalized s-manifolds and explores their polars and antipodal sets.
problem Understanding polars and antipodal sets in generalized s-manifolds.
method Introduced generalized s-manifolds and provided a method to construct them. Studied polars and antipodal sets.
result Extended results on compact symmetric spaces to generalized s-manifolds.
One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds M+,M− introduced and studied by Chen and Nagano in [Totally geodesic submanifolds of symm…
The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal equivariance conditions that one can impose on a linear bijection between observables tha…
In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Rieman…
The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Ja…
The notion of two-numbers of connected Riemannian manifolds was introduced about 35 years ago in [Un invariant geometrique riemannien, C. R. Acad. Sci. Paris Math. 295 (1982), 389--391] by B.-Y. Chen and T. Nagano. Later, two-numbers have been studied by a number of mathematicians and it was then proved that two-number…
The present article is the final part of a series on the classification of the totally geodesic submanifolds of the irreducible Riemannian symmetric spaces of rank 2. After this problem has been solved for the 2-Grassmannians in my previous papers cited in the present paper as [K1] and [K2], and for the space SU(3)/SO(…
Survey on geometry and topology of maximal antipodal sets.
problem Maximal antipodal sets on Riemannian manifolds.
method Comprehensive survey of existing research.
result Relation to various mathematical areas.
An explicit classification of simply connected compact homogeneous CR manifolds G/L of codimension one, with non-degenerate Levi form, is given. There are three classes of such manifolds: a) the standard CR homogeneous manifolds which are homogeneous S^1-bundles over a flag manifold F, with CR structure induced by an i…
The paper describes projective structures with torsion using formal frames and Thomas-Whitehead connections.
problem Characterizing projective structures with torsion.
method Using formal frames and Thomas-Whitehead connections.
result Fundamental theorem for Thomas-Whitehead connections regardless of torsion triviality.
The 2-rank of a compact Lie group G is the maximal possible rank of the elementary 2-subgroup Z2×...Z2 of G. The study of 2-ranks (and p-rank for any prime p) of compact Lie groups was initiated in 1953 by A. Borel and J.-P. Serre. Since then the 2-ranks of compact Lie groups h…
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
(1,1) non-L-space knots are foliar in 3D space.
problem Proving (1,1) non-L-space knots are foliar.
method Analyzing (1,1) non-L-space knots in S3 and lens spaces. result (1,1) non-L-space knots are persistently foliar.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
The distance function ϱ(p,q) (or d(p,q)) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over Rn, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
In a paper (math.DG/0403528) we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, …
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.
Study on convergence of transformed metric spaces as dimensions grow.
problem Conditions for convergence of transformed metric spaces.
method Clarifying conditions for convergence of transformed spaces from original sequence and vice versa.
result Spheres and projective spaces converge to Gaussian space and its quotient as dimensions increase.
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
New quasi space forms solve Thurston's geometrical space form problem.
problem Solving Thurston's geometrical space form problem.
method Introducing quasi space forms as non-real space forms with specific geometric properties.
result Quasi space forms offer a metrical, local geometrical solution to Thurston's problem.
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
Introduces new types of homogeneous spaces and their properties.
problem Defining and understanding new types of homogeneous spaces.
method Introducing and analyzing (strongly) (Θ-)discrete homogeneous spaces. result Discovers relationships between new and existing homogeneous space types.
This work describes compactifications of metric spaces and vector spaces using asymmetric norms.
problem Compactifying metric spaces and vector spaces using asymmetric norms.
method Nonstandard methods, ultrapowers of the spaces at hand.
result Polyhedral compactifications of vector spaces with stratified structure.
In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite …
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
problem Characterizing CAT(0) spaces with specific volume growth properties.
method Analyzing asymptotic topological regularity and volume growth.
result CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
Cantor Riemannium is a new type of space from holomorphic germs.
problem Defining a new type of space from holomorphic germs.
method Constructing the Cantor Riemannium by Borel monogenic continuation.
result The Cantor Riemannium is a metric, path connected, Gromov length space.
We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of symplect…
Study static Einstein-Maxwell space invariant by translation.
problem Classify static Einstein-Maxwell space invariant under translation.
method Analyze static Einstein-Maxwell space conformal to pseudo-Euclidean space.
result Complete classification of static Einstein-Maxwell space invariant under translation.
A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace …
The classification of G-spaces by Palais is refined for the case where the orbit space satisfies certain mild topological hypotheses. It is shown that when a sequence of such orbit spaces is "close" to a limit orbit space, in some suitable sense, within a larger ambient orbit space, the G-spaces in the tail of the sequ…
New spaces help connect manifold structures on equivariant Poincaré spaces.
problem Creating manifold structures on equivariant Poincaré spaces.
method Introducing semifree isovariant G-Poincaré spaces and gap conditions. result Space of isovariant structures on semifree G-Poincaré spaces is highly connected.