No abstract

Abstract. We classify extended Poincaré Lie super algebras and Lie algebras of any signature (p, q), that is Lie super algebras and Z 2 -graded Lie algebras g = g 0 + g 1 , where g 0 = so(V ) + V is the (generalized) Poincaré Lie algebra of the pseudo Euclidean vector space V = R p,q of signature (p, q) and g 1 = S is the spinor so(V )-module extended to a g 0 -module with kernel V . The remaining super commutators {g 1 , g 1 } (respectively, commutators [g 1 , g 1 ]) are defined by an so(V )-equivariant linear mappingDenote by P + (n, s) (respectively, P − (n, s)) the vector space of all such Lie super algebras (respectively, Lie algebras), where n = p + q = dim V and s = p − q is the signature. The description of P ± (n, s) reduces to the construction of all so(V )-invariant bilinear forms on S and to the calculation of three Z 2 -valued invariants for some of them.This calculation is based on a simple explicit model of an irreducible Clifford module S for the Clifford algebra Cl p,q of arbitrary signature (p, q). As a result of the classification, we obtain the numbers L ± (n, s) = dim P ± (n, s) of independent Lie super algebras and algebras, which take values 0,1,2,3,4 or 6. Due to Bott periodicity, L ± (n, s) may be considered as periodic functions with period 8 in each argument. They are invariant under the group Γ generated by the four reflections with respect to the axes n = −2, n = 2, s − 1 = −2 and s − 1 = 2. Moreover, the reflection (n, s) → (−n, s) with respect to the axis n = 0 interchanges L + and L − :

We study the geometry of multidimensional scalar 2 nd order PDEs (i.e. PDEs with n independent variables) with one unknown function, viewed as hypersurfaces E in the Lagrangian Grassmann bundle M (1) over a (2n + 1)-dimensional contact manifold (M, C). We develop the theory of characteristics of the equation E in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of E. After specifying the results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat in [11], i.e. MAEs of the formWe show that any MAE of the aforementioned class is associated with an n-dimensional subdistribution D of the contact distribution C, and viceversa. We characterize this Goursat-type equations together with its intermediate integrals in terms of their characteristics and give a criterion of local contact equivalence. Finally, we develop a method of solutions of a Cauchy problem, provided the existence of a suitable number of intermediate integrals.

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form \omega . This generalises Freed's definition of (affine) special K\"ahler manifolds. We also define projective versions of all these geometries. Our main result is an extrinsic realisation of all simply connected (affine or projective) special complex, symplectic and K\"ahler manifolds. We prove that the above three types of special geometry are completely solvable, in the sense that they are locally defined by free holomorphic data. In fact, any special complex manifold is locally realised as the image of a holomorphic 1-form \alpha : C^n \to T^* C^n. Such a realisation induces a canonical \nabla-parallel symplectic structure on M and any special symplectic manifold is locally obtained this way. Special K\"ahler manifolds are realised as complex Lagrangian submanifolds and correspond to closed forms \alpha. Finally, we discuss the natural geometric structures on the cotangent bundle of a special symplectic manifold, which generalise the hyper-K\"ahler structure on the cotangent bundle of a special K\"ahler manifold.Comment: 24 pages, latex, section 3 revised (v2), modified Abstract and Introduction, version to appear in J. Geom. Phy

Abstract. A geodesic in a Riemannian homogeneous manifold (M = G/K, g)is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of the Lie group G. We investigate G-invariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when M = G/K is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group G. We use an important invariant of a flag manifold M = G/K, its T -root system, to give a simple necessary condition that M admits a non-standard G-invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds M = G/K of a simple Lie group G, only the manifold Com(R 2 +2 ) = SO(2 + 1)/U ( ) of complex structures in R 2 +2 , and the complex projective space CP 2 −1 = Sp( )/U (1) · Sp( − 1) admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only G-invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra g of G). According to F. Podestà and G. Thorbergsson (2003), these manifolds are the only nonHermitian symmetric flag manifolds with coisotropic action of the stabilizer.