We define a new version of the exterior derivative on the basic forms of a Riemannian foliation to obtain a new form of basic cohomology that satisfies Poincaré duality in the transversally orientable case. We use this twisted basic cohomology to show relationships between curvature, tautness, and vanishing of the basic Euler characteristic and basic signature.

We apply the Atiyah–Singer index theorem and tensor products of elliptic complexes to cohomology transitive Lie algebroids. prove that Euler characteristic a representation algebroid $A$ over compact manifold $M$ vanishes unless $A=TM$, general Künneth formula. As applications, we give short proof vanishing result for principal bundle calculated using invariant differential forms, show certain ...

This article treats connections on ber bundles B = P G F that are induced by a connection 1-form on the associated principal bundle P. Using horizontal lifts of vector elds it is shown which combinations of diierential forms on the ber F and on P canonically deene diierential forms on B. Local representations for these forms involving the gauge potentials and elds of the connection are given an...

Bryant and Salamon constructed metrics with holonomy G2 and Spin(7) on spin bundles of 3-dimensional space forms, and spin bundles and bundles of anti-self-dual 2-forms on self-dual Einstein 4-manifolds [BrS]. Since, apart from holonomy, the construction of integrable G2(respectively Spin(7)) structures amounts to finding differential 3(4)forms of generic type on 7(8) manifolds satisfying appro...

We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the half-line (0,∞) and the interval (−1, 1). This process is related to the square root of one-dimensional Laplacian A = − √ − d dx with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula ...

For any finite simple graphG = (V,E), the discrete Dirac operator D = d+d∗ and the Laplace-Beltrami operator L = dd∗+d∗d = (d+d∗)2 = D2 on the exterior algebra bundle Ω = ⊕Ωk are finite v × v matrices, where dim(Ω) = v = ∑ k vk, with vk = dim(Ωk) denoting the cardinality of the set Gk of complete subgraphs Kk of G. We prove the McKean-Singer formula χ(G) = str(e−tL) which holds for any complex ...

The purpose of this paper is to enhance the conventional Komar integral asymptotically anti-de Sitter (AdS) black holes. In order do so, we first obtain a potential that linear combination usual with two third-order derivative terms generated by action d'Alembertian operator and exterior upon Killing vector. Then higher-order corrected extended Einstein gravity negative cosmological constant, y...

Exterior tomographic data are taken over lines outside a central region, and such data occur in the industrial nondestructive evaluation of large objects such as rockets. We explain, using microlocal analysis, which singularities are well reconstructed from exterior data, and we explain how this phenomenon is reflected in the singular value decomposition for the exterior transform [E.T. Quinto,...

Let R be a commutative ring. Unless indicated otherwise, all modules are R-modules and all tensor products are taken over R, so we abbreviate ⊗R to ⊗. A bilinear function out of M1 × M2 turns into a linear function out of the tensor product M1 ⊗ M2. In a similar way, a multilinear function out of M1 × · · · ×Mk turns into a linear function out of the k-fold tensor product M1 ⊗ · · · ⊗Mk. We wil...