In this paper it is shown that a Lanczos potential for the Weyl curvature tensor does not exist for all spaces of dimension n ≥ 7. Whether there exists a Lanczos potential [1] for Weyl curvature tensors in dimensions n > 4 has still not been determined. Although Lanczos's original proof [1] for existence was flawed [2], there have subsequently been complete proofs for existence in four dimensio...

We present some results in [9], a continuation of our earlier works [7] and [8]. One result is the existence and compactness of solutions to a fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds, and the other is a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations. Let (M, g) be an n−dimensional, ...

Our main results are: (1) The complex and Lagrangian points of a non-complex and nonLagrangian 2n-dimensional submanifold F :M →N , immersed with parallel mean curvature and with equal Kähler angles into a Kähler-Einstein manifold (N, J, g) of complex dimension 2n, are zeros of finite order of sin θ and cos θ respectively, where θ is the common J-Kähler angle. (2) If M is a Cayley submanifold o...

The notion of the Wick star-product is covariantly introduced for a general symplectic manifold equipped with two transverse polarisations. Along the lines of Fedosov method, the explicit procedure is given to construct the Wick symbols on the manifold. The cohomological obstruction is identified to the equivalence between the Wick star-product and the Fedosov one. In particular in the Kähler c...

Generalized composite fluxbrane solutions for a wide class of intersection rules are obtained. The solutions are defined on a manifold which contains a product of n Ricci-flat spaces M1 × . . . × Mn with 1-dimensional M1 . They are defined up to a set of functions Hs obeying non-linear differential equations equivalent to Toda-type equations with certain boundary conditions imposed. A conjectur...

A “conformal tensor” is constructed from the metric tensor gMN (or Vielbein e A M ) and is invariant against Weyl rescaling gMN → egMN (or eM → eeM ). Moreover, it vanishes if and only if the space is conformally flat, gMN = e ηMN (or e A M = eδ M ). In dimension four or greater the conformal tensor is the Weyl tensor. In three dimensions the Weyl tensor vanishes identically, while the Cotton t...

Some results on the properties of T -flat, quasiT -flat,  T -flat,  T -flat, T -semi-symmetric,  T Ricci recurrent and T - -recurrent LP-Sasakian manifolds are obtained. It is also proved that an LP-Sasakian manifold satisfying the condition T . 0 S  is an  -Einstein manifold. MSC 2000. 53C15, 53C25, 53C50, 53D15.

We study canonical paracontact connection on a para-Sasakian manifold. We prove that a Ricci-flat para-Sasakian manifold with respect to canonical paracontact connection is an η-Einstein manifold.We also investigate some properties of curvature tensor, conformal curvature tensor,W2curvature tensor, concircular curvature tensor, projective curvature tensor, and pseudo-projective curvature tensor...

In this note we define the Chern-Simons classes of a flat superconnection, D + L, on a complex Z/2Z-graded vector bundle E on a manifold such that D preserves the grading and L is an odd endomorphism of E. As an application, we obtain a definition of Chern-Simons classes of a (not necessarily flat) morphism between flat vector bundles on a smooth manifold. An application of Reznikov’s theorem s...

Let G be a 1-connected, almost-simple Lie group over a local field and S a subsemigroup of G with non-empty interior. The action of the regular hyperbolic elements in the interior of S on the flag manifold G/P and on the associated Euclidean building allows us to prove that the invariant control set exists and is unique. We also provide a characterization of the set of transitivity of the contr...