The algebra of parallel endomorphisms of a pseudo-Riemannian metric: semi-simple part

Abstract

Abstract. On a (pseudo-)Riemannian manifold (M, g), some fields of endomorphisms i.e. sections of End(T M) may be parallel for g. They form an associative algebra e, which is also the commutant of the holonomy group of g. As any associative algebra, e is the sum of its radical and of a semi-simple algebra s. Here we study s: it may be of eight different types, including the generic type s = R Id, and the Kähler and hyperkähler types s ≃ C and s ≃ H. This is a result on real, semi-simple algebras with involution. For each type, the corresponding set of germs of metrics is non-empty; we parametrise it. We give the constraints imposed to the Ricci curvature by parallel endomorphism fields.Keywords: Pseudo-Riemannian, Kähler, hyperkähler, parakähler metrics, holonomy group, parallel endomorphism, nilpotent endomorphism, commutant, Ricci curvature, real algebra with involution, semi-simple associative algebra.M.S.C. 2010: 53B30, 53C29, 16K20, 16W10 secondary 53B35, 53C10, 53C12, 15A21.We classify here the germs of (pseudo-)Riemannian metric, after the semi-simple part of their algebra of parallel endomorphism fields. Our motivation is the following.Motivation. A Kähler metric g on some manifold M may be defined as a Riemannian metric admitting an almost complex structure J which is parallel: DJ = 0 with D the LeviCivita connection of g. A natural question is to ask whether other fields of endomorphisms, i.e. sections of End(T M), may be parallel for a Riemannian metric. The answer is nearly immediate. First, one restricts the study to metrics that do not split into a non trivial Riemannian product, called here "indecomposable". Otherwise, any parallel endomorphism field is the direct sum of parallel such fields on each factor (considering as a unique factor the possible flat factor). Then a brief reasoning ensures that only three cases occur: g may be generic i.e. admit only the homotheties as parallel endomorphisms, be Kähler, or be hyperkähler i.e. admit two (hence three) anticommuting parallel complex structures. The brevity of this list is due to a simple fact: the action of the holonomy group H of an indecomposable Riemannian metric is irreducible i.e. does not stabilise any proper subspace. In particular, this compels any parallel endomorphism field to be of the form λ Id +µJ with J some parallel, skew adjoint almost complex structure. Now, such irreducibility fails in general for an indecomposable pseudo-Riemannian metric, so that a miscellany of other parallel endomorphism fields may appear. This gives rise to the question tackled here:

On the algebra of parallel endomorphisms of a pseudo-Riemannian metric

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On a (pseudo-)Riemannian manifold (M, g), some fields of endomorphisms i.e. sections of End(T M) may be parallel for g. They form an associative algebra e, which is also the commutant of the holonomy group of g. As any associative algebra, e is the sum of its radical and of a semi-simple algebra s. This s may be of eight different types, see [8]. Then, for any self adjoint nilpotent element N of the commutant of such an s in End(T M), the set of germs of metrics such that e ⊃ s ∪ {N } is non-empty.

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“…End(T m M) H = s ⊕ n with n := Rad(End(T m M) H ) a nilpotent ideal, its radical, and s ≃ End(T m M) H )/n a semi-simple subalgebra. See [8] for details.…”

“…Once again, only the semi-simple part s allows an exhaustive treatment. We provided it in [8]: s may be of eight different types, including the generic, Kähler and hyperkähler types s ≃ R, s ≃ C and s ≃ H, plus five other ones appearing only for pseudo-Riemannian metrics; see here Theorem 1.1 p. 5. On the contrary, no list of possible forms for n may presently be given.…”

“…[1] and its bibliography may be consulted. Besides, our geometric context does not seem to simplify significantly the algebraic nature of n. So we treat here a natural first step, the case where End(T m M) H contains: -one of the eight semi-simple algebras s ⊂ End(T m M) listed in [8], -and some given nilpotent endomorphism N not belonging to s.…”

“…Theorem Take s one of the eight algebras listed in Theorem 1.10 of [8], reproduced here in section 1 p. 5, and N any self adjoint nilpotent element of the commutant of s in End(T m M). The set of germs of metrics such that End(T m M) H contains s ∪ {N } i.e.…”