Neda Bokan

A Note on Osserman Lorentzian Manifolds

Let p be a point of a Lorentzian manifold M . We show that if M is spacelike Osserman at p , then M has constant sectional curvature at p ; similarly, if M is timelike Osserman at p , then M has constant sectional curvature at p . The reverse implications are immediate. The timelike case and 4-dimensional spacelike case were first studied in [ 3 ]; we use a different approach to this case.

Osserman pseudo-Riemannian manifolds of signature (2,2)

A pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Jf?x is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of J£jhas to have a triple zero, which is the other main result. An important step in the proof is based on Walkers study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable JacobToperators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds. 2000 Mathematics subject classification: primary 53B30, 53C50.

On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection

We give a complete decomposition of the space of curvature tensors with the symmetry properties as the curvature tensor associated with a symmetric connection of Riemannian manifold. We solve the problem under the action ofS0(n). The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for Riemannian manifolds with symmetric connection are given. The group of projective transformations of a Riemannian manifold and its subgroups are considered.

A note on the Osserman conjecture and isotropic covariant derivative of curvature

Let M be a Riemannian manifold with the Jacobi operator, which has constant eigenvalues, independent on the unit vector X ∈ TpM and the point p ∈ M . Osserman conjectured that these manifolds are flat or rankone locally symmetric spaces (∇R = 0). It is known that for a general pseudo-Riemannian manifold, the Osserman-type conjecture is not true and 4dimensional Kleinian Jordan-Osserman manifolds are curvature homogeneous. We show that the length of the first covariant derivative of the curvature tensor is isotropic, i.e. ‖∇R‖ = 0. For known examples of 4-dimensional Osserman manifolds of signature (−−++) we check also that ‖∇R‖ = 0. By the presentation of a class of examples we show that curvature homogeneity and ‖∇R‖ = 0 do not imply local homogeneity; in contrast to the situation in the Riemannian geometry, where it is unknown if the Osserman condition implies local homogeneity. §0. Introduction Let (M, g) be a 4-dimensional Kleinian (neutral) manifold, i.e. a pseudo-Riemannian manifold with a metric g of signature (− − ++). We denote its curvature tensor by R. The Jacobi operator RX : Y 7→ R(Y, X)X is a symmetric endomorphism of TpM and KX is its restriction to X⊥ in TpM . For Riemannian manifolds, Osserman [16], based on joint results with Sarnak [17], has conjectured that if the eigenvalues of the Jacobi operator KX are independent of the choice of unit vectors X ∈ TpM and of the choice p ∈ M , then either M is locally a rank-one symmetric space or M is flat. We have generalized in [3] the Osserman-type condition in the pseudo-Riemannian setup in terms of the Jordan form of KX , that is equivalent, especially for 4-dimensions, to the conditions in terms of the constancy of the minimal polynomial for KX . Namely, M is spacelike (resp. timelike) Jordan-Osserman at p if the Jordan form of KX is independent of X ∈ TpM , g(X, X) = 1 (resp. g(X, X) = −1). If M is spacelike (resp. timelike) Jordan-Osserman at every p ∈ M , one says M is pointwise spacelike (resp. timelike) Jordan-Osserman. If the Jordan form of KX is independent of p ∈ M , then M is spacelike (resp. timelike) Jordan-Osserman. Received by the editors November 6, 1997 and, in revised form, March 3, 1998. 1991 Mathematics Subject Classification. Primary 53B30, 53C50.

Foliation of a dynamically homogeneous neutral manifold

It is known that Riemannian and Lorentzian four-dimensional dynamically homogeneous manifolds are two-point homogeneous spaces. This is not true for signature (−−++) (neutral or Kleinian signature). In order to better understand their rich structure we study the geometry of nonsymmetric dynamically homogeneous spaces (types II and III): they admit autoparallel distributions and they are locally foliated by totally geodesic, flat, isotropic two-dimensional submanifolds. Moreover we characterize them locally in terms of the existence of an appropriate coordinate system (in the sense of A. G. Walker [Q. J. Math. 1, 69–79 (1950)]).

Geometry of differential operators on Weyl manifolds

We define natural operators of Laplace type for a Weyl manifold which transform conformally. We use the asymptotics of the heat equation for these operators to construct global invariants in Weyl geometry.

An inhomogeneous elliptic complex

We define a 3 term sequenceP of differential operators of mixed type; the first and third operators are 1st order while the second operator is 2nd order.P is always elliptic; it forms a complex ifM is einstein. It was first discussed by Gasqui.P is related to similar complexesC andG discussed by 02 Calabi and Gasqui-Goldschmidt. The index and equivariant index ofP vanish. In dimension 2,P=C⊗s whereS is of Dirac type;C and-S determine the same equivariant index. We study the heat equation asymptotics of the operators ofP; the associated Laplacians do not have scalar leading symbol.

Geometric Structures as Determined by the Volume of Generalized Geodesic Balls

Several authors have studied the Taylor expansion for the volume of geodesic balls under the exponential mapping of an analytic Riemannian manifold % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Holonomy, Geometry and Topology of Manifolds with Grassmann Structure

(M, {\cal G})

Pontrjagin forms, Chern Simons classes, Codazzi transformations, and affine hypersurfaces

. A more general structure % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!