Mappings preserving area 1 of triangles

This paper is concerned with the characterization of area-preserving mappings of real inner product spaces.     

Mappings preserving two hyperbolic distances

Suppose that X is the set of points of a hyperbolic geometry of finite or infinite dimension , and that is a fixed real number and N>1 a fixed integer. Let be a mapping such that for every if h(x, y)=, then h(f,(x),f,(y)) , and if h(x,y) = N, then h(f,(x),f,(y)) , where h,(p,q) designates the hyperbolic distance of p,q . Then f is an isometry of X. Note that there is no regularity assumption on f, like continuity or even differentiability. Moreover, we present an example showing that the assumption that one fixed distance > 0 is preserved does not characterize hyperbolic isometries. Received 17 February 2000.     

Why are surjective lineations of the Archimedean hyperbolic plane motions?

Positive L_ω1ω definitions of point-inequality and noncollinearity in terms of collinearity, which are valid in plane hyperbolic geometry over arbitrary Archimedean ordered Euclidean fields, provide a synthetic proof of the theorem stated in the title and first noticed to be a corollary of a result from [2] by R. Höfer [3].     

On the maps preserving the equality of distance

In this paper, in connection with the known result of Baker and Vogt, we get if preserves equality of distance with dimension X?2 and Y is strictly convex, the range of f contains a segment, then f is affine.     

Mappings preserving the idempotency of products of operators

We obtain a general form of a surjective (not assumed additive) mapping φ, preserving the nonzero idempotency of a certain product, being defined (a) on the algebra of all bounded linear operators B(X), where X is at least three-dimensional real or complex Banach space, (b) on the set of all rank-one idempotents in B(X) and (c) on the set of all idempotents in B(X). In any of the cases it turns out that φ is additive and either multiplicative or antimultiplicative.     

Mappings preserving spectra of products of matrices

Let be the set of complex matrices, and for every , let denote the spectrum of . For various types of products on , it is shown that a mapping satisfying for all has the form

for some invertible and scalar . The result covers the special cases of the usual product , the Jordan triple product , and the Jordan product . Similar results are obtained for Hermitian matrices.

     

Dynamical invariants of rigid motions on the hyperbolic plane

This paper is devoted to the investigation of the geometry of left invariant metrics on the isometry group of the hyperbolic plane determined by the kinetic energy of a rigid particle system.Dedicated to the memory of Professor A. M. Vasil'ev (1923–1987)     

CR-submanifolds of complex hyperbolic spaces satisfying a basic equality

The first author introduced a Riemannian invariant denoted by δ and proved in [4] that everyn-dimensional submanifold of the complex hyperbolicm-space ℂH ^m (4c) of constant holomorphic sectional curvature 4c<0 satisfies a basic inequality , whereH ² denotes the squared mean curvature of the submanifold. The main purpose of this paper is to completely classify properCR-submanifolds of complex hyperbolic spaces which satisfy the equality case of this inequality. Dedicated to Leopold Verstraelen on his fiftieth birthday     

Equality of pressures for diffeomorphisms preserving hyperbolic measures

For a diffeomorphism which preserves a hyperbolic measure the potential is studied. Various types of pressure of are introduced. It is shown that these pressures satisfy a corresponding variational principle. This research was supported by the grant EU FP6 ToK SPADE2. The author is grateful to IM PAN Warsaw for the hospitality and to C. Wolf for discussions about suitable concepts of pressure.     

Pairs of theonian triangles with common area

We study the properties of pairs of triangles with integer sides whose common area is the square of a natural number.     

Upper estimates for hyperbolic metrics on subdomains: the case of equality

It is the purpose of this paper to supplement our previous work concerning upper estimates for hyperbolic metrics on subdomains. Specifically, our interest is in determining when equality holds in the relevant inequality to clarify the situation that has failed to be explicitly mentioned before. The method of proof again involves a standard technique based on the classical Schwarz–Pick lemma.