Principal homogeneous spaces and group scheme extensions

On the Geometry of Principal Homogeneous Spaces

Let k be an algebraically closed field, let π : X → B be an elliptic surface defined over k, and let XK be the generic fiber of π, which is an elliptic curve defined over the field K = k(B), the function field of B. If f : Y → B is a genus one fibration locally isomorphic to X (in the étale topology on B), then Y corresponds to a principal homogeneous space YK over XK which is everywhere locall...

On the Geometry of Principal Homogeneous Spaces

Let B be a curve defined over an algebraically closed field k and let π: X → B be an elliptic surface with base curve B. We investigate the geometry of everywhere locally trivial principal homogeneous spaces for X, i.e., elements of the Tate-Shafarevich group of the generic fiber of π. If Y is such a principal homogeneous space of order n, we find strong restrictions on the Pn−1 bundle over B i...

Frames and Homogeneous Spaces

Let be a locally compact non?abelian group and be a compact subgroup of also let be a ?invariant measure on the homogeneous space . In this article, we extend the linear operator as a bounded surjective linear operator for all ?spaces with . As an application of this extension, we show that each frame for determines a frame for and each frame for arises from a frame in via...

Convolution and Homogeneous Spaces

Homogeneous Spaces of the Lorentz Group

We present a classification, up to isomorphisms, of all the homogeneous spaces of the Lorentz group with dimension lower than six. At the same time, we classify, up to conjugation, all the non-discrete closed subgroup of the Lorentz group and all the subalgebras of the Lorentz Lie algebra. We also study the covariant mappings between some pairs of homogeneous spaces. This exercise is done witho...