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4-dimensional anti-Kähler manifolds and Weyl curvature

Jaeman Kim (2006)

Czechoslovak Mathematical Journal

On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.

A geometry on the space of probabilities (II). Projective spaces and exponential families.

Henryk Gzyl, Lázaro Recht (2006)

Revista Matemática Iberoamericana

In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities (I). The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a...

A local characterization of affine holomorphic immersions with an anti-complex and ∇-parallel shape operator

Maria Robaszewska (2002)

Annales Polonici Mathematici

We study the complex hypersurfaces f : M ( n ) n + 1 which together with their transversal bundles have the property that around any point of M there exists a local section of the transversal bundle inducing a ∇-parallel anti-complex shape operator S. We give a class of examples of such hypersurfaces with an arbitrary rank of S from 1 to [n/2] and show that every such hypersurface with positive type number and S ≠ 0 is locally of this kind, modulo an affine isomorphism of n + 1 .

Asymptotic behaviour and the moduli space of doubly-periodic instantons

Olivier Biquard, Marcos Jardim (2001)

Journal of the European Mathematical Society

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line , with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to T × 1 . The converse statement is also true, namely a holomorphic bundle on T × 1 which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton....

Complex geodesics and isometries in Cartan domains of type four

Edoardo Vesentini (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Holomorphic maps of Cartan domains of type four preserving the supports of complex geodesics are characterized, providing, in particular, a new description of holomorphic isometries.

Complex sprays and complex curves.

Colum Watt (2001)

Revista Matemática Complutense

After defining what is meant by a complex spray X on a complex manifold M, we introduce the notion of a spray complex curve associated to X. Several equivalent formulations are derived and we give necessary and sufficient conditions for M to admit spray complex curves for X through each point and in each direction. Refinements of this result are then derived for some special cases, for example when X is the horizontal radial vector field associated to a complex Finsler metric.

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