{
  "id": "1207.6544",
  "version": "v6",
  "published": "2012-07-27T13:46:29.000Z",
  "updated": "2015-05-19T13:54:08.000Z",
  "title": "The algebra of parallel endomorphisms of a germ of pseudo-Riemannian metric",
  "authors": [
    "Charles Boubel"
  ],
  "comment": "47 pages. This version is only a part of the first version of this preprint. The other part is now published separately, see arXiv:1402.6642. Here some typos are corrected; some statements, remarks and tables are made more concise",
  "journal": "Journal of Differential Geometry 99 issue 1, 77-123, January 2015",
  "categories": [
    "math.DG"
  ],
  "abstract": "On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e. sections of End(TM) may be parallel for g. They form an associative algebra A, which is also the commutant of the holonomy group of g. As any associative algebra, A is the sum of its radical and of a semi-simple algebra S. We show in arXiv:1402.6642 that S may be of eight different types, including the generic type S=R.Id, and the K\\\"ahler and hyperk\\\"ahler types where S is respectively isomorphic to the complex field C or to the quaternions H. We show here that for any self adjoint nilpotent element N of the commutant of such an S in End(TM), the set of germs of metrics such that A contains S and {N} is non-empty. We parametrise it. Generically, the holonomy algebra of those metrics is the full commutant of $S\\cup\\{N\\}$ in O(g). Apart from some \"degenerate\" cases, the algebra A is then $S \\oplus (N)$, 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 describes totally the local situation when the radical of A is principal and consists of self adjoint elements. We add a glimpse on the case where this radical is not principal.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2013-12-13T17:00:24.000Z",
      "title": "The algebra of the parallel endomorphisms of a germ of pseudo-Riemannian metric",
      "abstract": "On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e. sections of End(TM) may be parallel for g. They form an associative algebra A, which is also the commutant of the holonomy group of g. As any associative algebra, A is the sum of its radical and of a semi-simple algebra S. We show the following: S may be of eight different types, including the generic type S=R.Id, and the K\\\"ahler and hyperk\\\"ahler types where S is respectively isomorphic to the complex field C or to the quaternions H. This is a result on real, semi-simple algebras with involution. Then, for any self adjoint nilpotent element N of the commutant of such an S in End(TM), the set of germs of metrics such that A contains S and {N} is non-empty. We parametrise it. Generically, the holonomy algebra of those metrics is the full commutant of $S\\cup\\{N\\}$ in O(g). Apart from some ``degenerate'' cases, the algebra A is then $S \\oplus (N)$, 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 describes totally the local situation when the radical of A is principal and consists of self adjoint elements. We add a glimpse on the case where this radical is not principal, and give the constraints imposed to the Ricci curvature when A is not reduced to R.Id.",
      "comment": "53 pages. Some typos corrected; some statements, remarks and tables made more concise",
      "journal": null,
      "doi": null
    },
    {
      "version": "v6",
      "updated": "2015-05-19T13:54:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53B30",
      "53C29",
      "16K20",
      "16W10",
      "53B35",
      "53C10",
      "53C12",
      "15A21"
    ],
    "keywords": [
      "pseudo-riemannian metric",
      "parallel endomorphisms",
      "self adjoint nilpotent element",
      "semi-simple algebra",
      "self adjoint elements"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.6544B"
    }
  }
}