On the algebra of parallel endomorphisms of a pseudo-Riemannian metric Article Swipe
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· 2015
· Open Access
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· DOI: https://doi.org/10.4310/jdg/1418345538
· OA: W2115658648
On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e.\nsections of End(TM) may be parallel for g. They form an associative algebra A,\nwhich is also the commutant of the holonomy group of g. As any associative\nalgebra, A is the sum of its radical and of a semi-simple algebra S. We show in\narXiv:1402.6642 that S may be of eight different types, including the generic\ntype S=R.Id, and the K\\"ahler and hyperk\\"ahler types where S is respectively\nisomorphic to the complex field C or to the quaternions H. We show here that\nfor any self adjoint nilpotent element N of the commutant of such an S in\nEnd(TM), the set of germs of metrics such that A contains S and {N} is\nnon-empty. We parametrise it. Generically, the holonomy algebra of those\nmetrics is the full commutant of $S\\cup\\{N\\}$ in O(g). Apart from some\n"degenerate" cases, the algebra A is then $S \\oplus (N)$, where (N) is the\nideal spanned by N. To prove it, we introduce an analogy with complex\nDifferential Calculus, the ring R[X]/(X^n) replacing the field C. This\ndescribes totally the local situation when the radical of A is principal and\nconsists of self adjoint elements. We add a glimpse on the case where this\nradical is not principal.\n
