Torus actions on homotopy complex projective spaces

We prove that an effective action of a torus T on a homotopy P^m is linear if m < 4 rk(T) – 1. Examples show that the bound is optimal. Combining this with a theorem of Hattori we conclude that the total Pontrjagin class of such a manifold is given by the usual formula (1 + x²)^{m + 1}.The second named author is an Alfred P. Sloan Research Fellow and was partly supported by an NSF grant.in final form: 10 May 2003     

On canonical embeddings of complex projective spaces in real projective spaces

In this paper, we construct a natural embedding \(\sigma :\mathbb{C}P_\mathbb{R}^{n} \to \mathbb{R}P^{n^2 + 2n} \) of the complex projective space ?P ⁿ considered as a 2n-dimensional, real-analytic manifold in the real projective space \(\mathbb{R}P^{n^2 + 2n} \). The image of the embedding σ is called the ?P ⁿ-surface. To construct the embedding, we consider two equivalent approaches. The first approach is based on properties of holomorphic bivectors in the realification of a complex vector space. This approach allows one to prove that a ?P-surface is a flat section of a Grassman manifold. In the second approach, we use the adjoint representation of the Lie group U(n + 1) and the canonical decomposition of the Lie algebra u(n). This approach allows one to state a gemetric characterization of the canonical decomposition of the Lie algebra u(n). Moreover, we study properties of the embedding constructed. We prove that this embedding determines the canonical Kähler structure on ?P _? ⁿ . In particular, the Fubini-Study metric is exactly the first fundamental form of the embedding and the complex structure on ?P _? ⁿ is completely defined by its second fundamental form; therefore, this embedding is said to be canonical. Moreover, we describe invariant and anti-invariant completely geodesic submanifolds of the complex projective space.     

Comparison of volumes of convex bodies in real, complex, and quaternionic spaces

The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in Rn with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n?4 and negative if n>4. The same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning. We give unified analysis of this circle of problems in real, complex, and quaternionic n-dimensional spaces. All cases are treated simultaneously. In particular, we show that the Busemann-Petty problem in the quaternionic n-dimensional space has an affirmative answer if and only if n=2. The method relies on the properties of cosine transforms on the unit sphere. We discuss possible generalizations.     

Codimension reduction for real submanifolds of quaternionic hyperbolic spaces

We prove a reduction theorem of codimension for real submanifolds of quaternionic hyperbolic spaces as a quaternionic analogue corresponding to those in Cecil [2], Erbacher [4], Kawamoto [8], Kwon and the second author [10] and Okumura [13].     

Entropy for semigroup actions on general topological spaces

This paper introduces both notions of topological entropy and invariance entropy for semigroup actions on general topological spaces. We use the concept of admissible family of open coverings to extending and studying the notions of Adler–Konheim–McAndrew topological entropy, Bowen topological entropy, and invariance entropy to the general theory of topological dynamics.     

Reiteration theorems for real interpolation and approximation spaces

Summary In the present paper we study a general form of Peetre's J- and K-methods of interpolation. Special emphasis is given to the equivalence theorem for J- and K-spaces and to reiteration theorems.     

Insertion and extension theorems for lattice-valued functions on preordered topological spaces

In this paper, the possibility of inserting an increasing continuous lattice-valued function between two comparable semicontinuous functions is studied. First, we obtain a sufficient condition for such insertion, then we get new characterizations of several classes of preordered topological spaces, among them normally preordered and extremally preorder-disconnected spaces. Conditions for the continuous and increasing extension of lattice-valued maps of the same type defined on closed (resp. open) sets are also investigated.     

Partial category actions on sets and topological spaces

We introduce (continuous) partial category actions on sets (topological spaces) and show that each such action admits a universal globalization. Thereby, we obtain a simultaneous generalization of corresponding results for groups, by Abadie, and Kellendonk and Lawson, and for monoids, by Megrelishvili and Schröder. We apply this result to the special case of partial groupoid actions where we obtain a sharpening of a result by Gilbert, concerning ordered groupoids, in the sense that mediating functions between universal globalizations always are injective.