Curvature tensors whose Jacobi or Szabo operator is nilpotent on null vectors.pdf

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Curvature tensors whose Jacobi or Szabo operator is nilpotent on null vectors.pdf

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Curvature tensors whose Jacobi or Szabo operator is nilpotent on null vectors

a r X i v : m a t h / 0 2 0 5 0 7 4 v 1 [ m a t h .D G ] 7 M a y 2 0 0 2 Curvature tensors whose Jacobi or Szabo? operator is nilpotent on null vectors Peter Gilkey? and Iva Stavrov February 8, 2008 Abstract We show that any k Osserman Lorentzian algebraic curvature tensor has constant sectional curvature and give an elementary proof that any local 2 point homogeneous Lorentzian manifold has constant sectional curvature. We also show that a Szabo? Lorentzian covariant derivative algebraic curvature tensor vanishes. 1 Introduction Let V be a vector space equipped with a symmetric inner product of signature (p, q) and dimension m = p+ q ≥ 3. V is said to be Riemannian if p = 0 and Lorentzian if p = 1. A 4 tensor R ∈ ?4V ? is said to be an algebraic curvature tensor if R has the symmetries of the curvature of the Levi-Civita connection: R(x, y, z, w) = R(z, w, x, y) = ?R(y, x, z, w), and R(x, y, z, w) +R(y, z, x, w) +R(z, x, y, w) = 0. A 5 tensor, which we denote symbolically by ?R ∈ ?5V ?, is said to be a covariant derivative algebraic curvature tensor if ?R has the symmetries of the covariant derivative of the curvature of the Levi-Civita connection: ?R(a, b, c, d; e) = ??R(b, a, c, d; e) = ?R(c, d, a, b; e), ?R(a, b, c, d; e) +?R(a, c, d, b; e) +?R(a, d, b, c; e) = 0, and ?R(a, b, c, d; e) +?R(a, b, d, e; c) +?R(a, b, e, c; d) = 0. The Jacobi operator JR(x) and the Szabo? operator SR(x) are the symmetric linear operators on V defined by: (JR(x)y, w) = R(y, x, x, w) and (S?R(x)y, w) = ?R(y, x, x, w;x). ?Research partially supported by the NSF (USA) and the MPI (Leipzig) 2000 Mathematics Subject Classification. Primary 53B20 Key words and phrases. Algebraic curvature tensor, nilpotent Jacobi operator, nilpotent Szabo? operator, Lorentzian geometry, null vector 1 In Section 2, we study the geometry of the Jacobi operator. Let k be an index 1 ≤ k ≤ m ? 1 and let {e1, ..., ek} be an orthonormal basis for a non- degenerate k dimensional subspace σ ? V . The higher