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

/pdf/a-class-of-almost-contact-riemannian-manifolds-1yxyggdui1.pdf

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 consider g-natural metrics on the tangent bundle of a Riemannian manifold together with the almost complex structure which reverses the horizontal and vertical subspaces. This narrows the class of g-natural metrics to metrics conformally equivalent to the Sasaki metric on the tangent bundle with a restriction on the conformal factor. We then show that such a g-natural almost Hermitian structure is Bochner flat if and only if it is conformally equivalent to the Sasaki metric when the base manifold is flat and with the same restriction on the conformal factor.

Bochner flatness of tangent bundles with g-natural almost Hermitian metrics

In this lecture we will discuss various generalizations of the catenoid and the helicoid as well as related differential geometric notions including minimality, quasi-umbilicity and conformal flatness.

Generalizations of the Catenoid and the Helicoid

Sasakian space forms

Considering odd-dimensional complex projective space as a
 complex contact manifold, one may ask which of the Calabi (Veronese) imbeddings can be positioned by a
 holomorphic congruence as integral submanifolds of the complex contact structure. It is first shown
 that when the first normal space is the whole normal space, this is impossible. It is also shown to be
 impossibile for a Calabi surface (complex dimension 2) in complex projective space of dimension 9
 where one has both a first and second normal space. However when the complex dimension of the
 submanifold is odd and the whole normal space consists of the first and second normal spaces, then
 there is a holomorphic congruence positioning the Calabi imbedding as an integral submanifold of the
 complex contact structure.

/pdf/the-calabi-veronese-imbeddings-as-integral-submanifolds-of-3sbz4mcban.pdf

The Calabi (Veronese) imbeddings as integral submanifolds of CP2n+1

In [1] B.-Y. Chen proved that if ~o:E\" ~ E\" is a conformal transformation of Euclidean space and M is a compact flat surface in E\" such that M = ~o(M) is also a compact flat surface, then there exists a similarity of E\" which also maps M to ~t. Of course, since a conformal map of E\"(n >/3) is the product of a similarity and an inversion and similarities preserve flatness, it is enough to prove this for a single inversion. Chen then shows that M must lie in a sphere concentric with the sphere of inversion. As a simple counterexample in the non-compact case, consider M to be an irregular patch on a cone with vertex at the center of inversion. The inversion then maps M to another such patch )~t, but there is no similarity mapping M to h4. In this paper we prove that the theorem of Chen and this example describe the general behavior of a conformal map mapping a flat submanifold of dimension ~> 3 to another such submanifold. Moreover the phenomena are essentially local in nature.

David E. Blair

Papers

Bochner flatness of tangent bundles with g-natural almost Hermitian metrics

Generalizations of the Catenoid and the Helicoid

Sasakian space forms

The Calabi (Veronese) imbeddings as integral submanifolds of CP2n+1

On conformal images of flat submanifolds