We introduce the notion of $\mathcal{D}$-homothetic bi-warping and starting from a Sasakian manifold $M$, we construct a family of Kählerian structures on the product $\mathbb{R} \times M.$ After, we investigate conditions on the product of a cosymplectic or Kenmotsu manifold and the real line to be a family of conformal Kähler manifolds. We construct several examples.

This work provides an algorithmic procedure for finding the pair of elements in the spin group which map to a given matrix in the special orthogonal group of order five or six. This is achieved by first solving the problem when the special orthogonal matrix is a Givens rotation, and then exploiting the fact that the covering maps are group homomorphisms and that any special orthogonal matrix can be explicitly decomposed into a product of Givens rotations. For this purpose systems of quadratic equations in several variables have to be solved symbolically. The resulting solution display a transparent dependency on the entries of the Givens matrices.

Here we present some new equations which we call Yang-Mills-Proca equations (or generalized Proca equations). This system of equations is a generalization of Proca equation and Yang-Mills equations and it is not gauge invariant. We present a number of constant solutions of this system of equations in the case of arbitrary Lie algebra. We consider in detail the case when this Lie algebra is a Clifford or a Grassmann algebra and derive solutions of Yang-Mills equations in the form of perturbation theory series near the constant solution.

In this paper we consider some Lie groups in complexified Clifford algebras. Using relations between operations of conjugation in Clifford algebras and matrix operations we prove isomorphisms between these groups and classical matrix groups (symplectic, orthogonal, linear, unitary) in the cases of arbitrary dimension and arbitrary signature. Also we obtain isomorphisms of corresponding Lie algebras which are direct sums of subspaces of quaternion types. Spin group is a subgroup of all considered groups and it coincides with one of them in the cases $n\leq 5$. We present classical matrix Lie groups that contain spin group in the case of arbitrary dimension.