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

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. 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. We parametrize it. Generically, the holonomy algebra of those metrics is the full commutant o(g) s∪{N } and then, apart from some "degenerate" cases, e = s ⊕ (N ), where (N ) is the ideal spanned by N . To prove it, we introduce an analogy with complex Differential Calculus, the ring R[X]/(X n ) replacing the field C. This treats the case where the radical of e is principal and consists of self adjoint elements. We add a glimpse on the case where this radical is not principal.

Degree of mobility for metrics of Lorentzian signature and parallel (0,2)-tensor fields on cone manifolds

“…As it was known already to Levi-Civita [15], two connections ∇ = Γ If ∇ and∇ related by (4) are Levi-Civita connections of metrics g andḡ, then one can find explicitly (following Levi-Civita [15]) a function φ on the manifold such that its differential φ ,i coincides with the 1-form φ i : indeed, contracting (4) with respect to i and j, we obtainΓ …”

“…The next two examples show that the assumptions in Section 3.3.2 thatĝ is a cone metric and that L has at most two nontrivial Jordan blocks are important. The first example is based on the description of nilpotent parallel symmetric (0, 2)-tensor fields due to Solodovnikov [12] and Boubel [4]. In order to produce the second example, we applied the construction from [21, Theorem 3.3] to g and L from the first example.…”

“…Thus, g(RL·, ·) ≡ 0 implyingRL = 0 as we claimed. The first example is based on the description of nilpotent parallel symmetric (0, 2)-tensor fields due to Kruchkovich and Solodovnikov [14] and Boubel [3]. In order to produce the second example, we applied the construction from [22, Theorem 3.3] to g and L from the first example.…”

Finite-dimensional integrable systems: A collection of research problems

“…The first one is a description of all possible algebras of parallel endomorphisms obtained by C. Boubel in [17]. The centralisers of these algebras are exactly the non-standard holonomy groups mentioned in Problem 24.…”

Argument Shift Method and Sectional Operators: Applications to Differential Geometry

“…For pseudo-Riemannian metrics, this problem turned out to be much more difficult. For the most important cases, local forms for g andḡ were obtained in [2,34,52], but the final solution has been obtained only recently [13,12,16,17].…”