Recently S. Tanno has classified connected almostcontact Riemannian manifolds whose automorphism groups have themaximum dimension [9]. In his classification table the almost contactRiemannian manifolds are divided into three classes: (1) homogeneousnormal contact Riemannian manifolds with constant 0-holomorphic sec-tional curvature if the sectional curvature for 2-planes which contain

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A class of almost contact riemannian manifolds

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

We give a description of all complete simply connected Riemannian manifolds carrying real Killing spinors. Furthermore, we present a construction method for manifolds with the exceptional holonomy groupsG
2 and Spin(7).

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104253076

Real Killing spinors and holonomy

http://www.dcs.gla.ac.uk/~muffy/papers/calder-kolberg-magill-reiff.pdf

Feature interaction: a critical review and considered forecast

We analyse the geometrical structure of supersymmetric solutions of type II supergravity of the form Open image in new window1,9−n×Mn with non-trivial NS flux and dilaton. Solutions of this type arise naturally as the near-horizon limits of wrapped NS fivebrane geometries. We concentrate on the case d=7, preserving two or four supersymmetries, corresponding to branes wrapped on associative or SLAG three-cycles. Given the existence of Killing spinors, we show that M7 admits a G2-structure or an SU(3)-structure, respectively, of specific type. We also prove the converse result. We use the existence of these geometric structures as a new technique to derive some known and new explicit solutions, as well as a simple theorem implying that we have vanishing NS three-form and constant dilaton whenever M7 is compact with no boundary. The analysis extends simply to other type II examples and also to type I supergravity.

G-Structures and Wrapped NS5-Branes

We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold with decomposable ϕ is a Calabi-Eckmann manifold or the Riemannian product of a sphere and . We show that a complete, simply connected, normal metric contact pair manifold with decomposable ϕ, such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, is the product of globally ϕ-symmetric spaces or the product of a globally ϕ-symmetric space and . Moreover in the first case the manifold fibers over a locally symmetric space endowed with a symplectic pair.

Symmetry in the Geometry of Metric Contact Pairs

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On the zeros of a conformal vector field

A sphere of dimension 4n+3 admits three Sasakian structures and it is natural to ask if a submanifold can be an integral submanifold for more than one of the contact structures. In the 7-sphere it is possible to have curves which are Legendre curves for all three contact structures and there are 2 and 3-dimensional submanifolds which are integral submanifolds of two of the contact structures. One of the results here is that if a 3-dimensional submanifold is an integral submanifold of one of the Sasakian structures and invariant with respect to another, it is an integral submanifold of the remaining structure and is a principal circle bundle over a holmophic Legendre curve in complex projective 3-space.

On the Geometry of the 7-Sphere

We prove that a complex (κ, µ)-space with κ < 1 is a locally homogeneous complex contact metric manifold. Also, a complex (κ, µ)-space has either κ = 1 or is GH-locally symmetric.

Homogeneity and local symmetry of complex (κ, µ)-spaces

A submanifold Mnr of Minkowski space \(\mathbb{E}_1^m \) is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of \(\mathbb{E}_1^m \) to the tangent space of Mnr at every point of Mnr. In this paper we completely classify hypersurfaces of restricted type in \(\mathbb{E}_1^{n + 1} \). More precisely, we prove that a hypersurface of \(\mathbb{E}_1^m \) is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: Sk × \(\mathbb{E}_1^{n - k} \), Sk1 × \(\mathbb{E}^{n - k} \), Hk × \(\mathbb{E}^{n - k} \), Sn1, Hn, with 1≤k≤n−1, or an open part of a cylinder on a plane curve of restricted type.

David E. Blair

Papers

Symmetry in the Geometry of Metric Contact Pairs

On the zeros of a conformal vector field

On the Geometry of the 7-Sphere

Homogeneity and local symmetry of complex (κ, µ)-spaces

Hypersurfaces of restricted type in Minkowski space