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Projective relatedness and conformal flatness

100%
Hall G.
Open Mathematics
|
2012
|
tom 10
|
nr 5
1763-1770
EN
This paper discusses the connection between projective relatedness and conformal flatness for 4-dimensional manifolds admitting a metric of signature (+,+,+,+) or (+,+,+,−). It is shown that if one of the manifolds is conformally flat and not of the most general holonomy type for that signature then, in general, the connections of the manifolds involved are the same and the second manifold is also conformally flat. Counterexamples are provided which place limitations on the potential strengthening of the results.
2
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Discrete Laplace cycles of period four

76%
Schröcker H.-P.
Open Mathematics
|
2012
|
tom 10
|
nr 2
426-439
EN
We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of corresponding points form a discrete W-congruence. We derive some properties of discrete Laplace cycles of period four and describe two explicit methods for their construction.
3
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Differential geometry of grassmannians and the Plücker map

76%
Anan’in S., Grossi C.
Open Mathematics
|
2012
|
tom 10
|
nr 3
873-884
EN
Using the Plücker map between grassmannians, we study basic aspects of classic grassmannian geometries. For ‘hyperbolic’ grassmannian geometries, we prove some facts (for instance, that the Plücker map is a minimal isometric embedding) that were previously known in the ‘elliptic’ case.
4
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Essential Killing fields of parabolic geometries: projective and conformal structures

64%
Čap A., Melnick K.
Open Mathematics
|
2013
|
tom 11
|
nr 12
2053-2061
EN
We use the general theory developed in our article [Čap A., Melnick K., Essential Killing fields of parabolic geometries, Indiana Univ. Math. J. (in press)], in the setting of parabolic geometries to reprove known results on special infinitesimal automorphisms of projective and conformal geometries.
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