# transformation group

## Non-reductive Homogeneous Spaces Not Admitting Normal Connections

The purpose of the work is the classification of three-dimensional non-reductive homogeneous spacesnot admitting normal connections, affine connections, their torsion tensors, curvature and holonomy algebras.The object of investigation arepointed-non-reductive spaces and connections on them. The basic notions, such as the isotropically-faithful pair, reductive space, afne connection, curvature tensor and torsion tensor, holonomy algebra and normal connection are defined.

## Connections of Nonzero Curvature on Three-dimensional Non-reductive Spaces

When a homogeneous space admits an invariant affine connection? If there exists at least one invariant connection then the space is isotropy-faithful, but the isotropy-faithfulness is not sufficient for the space in order to have invariantconnections. If a homogeneousspace is reductive, then the space admits an invariant connection.Thepurposeoftheworkisadescriptionofthree-dimensionalnon-reductivehomogeneousspaces, admitting invariant affine connections of nonzero curvature only, and the affine connections, curvature and torsion tensors.

## Three-dimensional Homogeneous Spaces, Not Admitting Invariant Connections

The purpose of the work is the classification of three-dimensional isotropy-faithful homogeneous spaces, not admitting invariant connections. The local classification of homogeneous spaces is equivalent to the description of effective pairs of Lie algebras. If there exists at least one invariant connection then the space is isotropy-faithful, but the isotropy-faithfulness is not sufficient for the space in order to have invariant connections.

## On a form of the first variation of the action integral over a varied domain

Field theories of the continuum mechanics and physics based on the least action principle are considered in a unified framework. Variation of the action integral in the least action principle corresponds variations of physical fields while space-time coordinates are not varied. However notion of the action invariance, theory of variational symmetries of action and conservation laws require a wider variation procedure including variations of the space-time coordinates.