Transitive conformal holonomy groups

Jesse Alt

Open Mathematics (2012)

  • Volume: 10, Issue: 5, page 1710-1720
  • ISSN: 2391-5455

Abstract

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For (M, [g]) a conformal manifold of signature (p, q) and dimension at least three, the conformal holonomy group Hol(M, [g]) ⊂ O(p + 1, q + 1) is an invariant induced by the canonical Cartan geometry of (M, [g]). We give a description of all possible connected conformal holonomy groups which act transitively on the Möbius sphere S p,q, the homogeneous model space for conformal structures of signature (p, q). The main part of this description is a list of all such groups which also act irreducibly on ℝp+1,q+1. For the rest, we show that they must be compact and act decomposably on ℝp+1,q+1, in particular, by known facts about conformal holonomy the conformal class [g] must contain a metric which is either Einstein (if p = 0 or q = 0) or locally isometric to a so-called special Einstein product.

How to cite

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Jesse Alt. "Transitive conformal holonomy groups." Open Mathematics 10.5 (2012): 1710-1720. <http://eudml.org/doc/269749>.

@article{JesseAlt2012,
abstract = {For (M, [g]) a conformal manifold of signature (p, q) and dimension at least three, the conformal holonomy group Hol(M, [g]) ⊂ O(p + 1, q + 1) is an invariant induced by the canonical Cartan geometry of (M, [g]). We give a description of all possible connected conformal holonomy groups which act transitively on the Möbius sphere S p,q, the homogeneous model space for conformal structures of signature (p, q). The main part of this description is a list of all such groups which also act irreducibly on ℝp+1,q+1. For the rest, we show that they must be compact and act decomposably on ℝp+1,q+1, in particular, by known facts about conformal holonomy the conformal class [g] must contain a metric which is either Einstein (if p = 0 or q = 0) or locally isometric to a so-called special Einstein product.},
author = {Jesse Alt},
journal = {Open Mathematics},
keywords = {Conformal holonomy; Transitive group actions; conformal holonomy; transitive group action},
language = {eng},
number = {5},
pages = {1710-1720},
title = {Transitive conformal holonomy groups},
url = {http://eudml.org/doc/269749},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Jesse Alt
TI - Transitive conformal holonomy groups
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1710
EP - 1720
AB - For (M, [g]) a conformal manifold of signature (p, q) and dimension at least three, the conformal holonomy group Hol(M, [g]) ⊂ O(p + 1, q + 1) is an invariant induced by the canonical Cartan geometry of (M, [g]). We give a description of all possible connected conformal holonomy groups which act transitively on the Möbius sphere S p,q, the homogeneous model space for conformal structures of signature (p, q). The main part of this description is a list of all such groups which also act irreducibly on ℝp+1,q+1. For the rest, we show that they must be compact and act decomposably on ℝp+1,q+1, in particular, by known facts about conformal holonomy the conformal class [g] must contain a metric which is either Einstein (if p = 0 or q = 0) or locally isometric to a so-called special Einstein product.
LA - eng
KW - Conformal holonomy; Transitive group actions; conformal holonomy; transitive group action
UR - http://eudml.org/doc/269749
ER -

References

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