We formulate the equivalence problem, in the sense of É. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety $Z$, a family of minimal rational curves with $Z$-isotrivial varieties of minimal rational tangents...

The author constructs the gauged Skyrme model by introducing the skyrmion bundle as follows: instead of considering maps $U:M\to {\text{SU}}_{N_F}$ he thinks of the meson fields as of global sections in a bundle $B(M,{\text{SU}}_{N_F},G)=P(M,G){\times }_G{\text{SU}}_{N_F}$. For calculations within the skyrmion bundle the author introduces by means of the so-called equivariant cohomology an analogue of the topological charge and the Wess-Zumino term. The final result of this paper is the following Theorem. For the skyrmion bundle with $N_F\le 6$, one has $$H^{*}\left(EG{\times }_G{\text{SU}}_{N_F}\right)\cong H^{*}{\left({\text{SU}}_{N_F}\right)}^G\cong \text{S}\left({\underline{G}}^{*}\right)\otimes H^{*}\left({\text{SU}}_{N_F}\right)\cong H^{*}\left(BG\right)\otimes H^{*}\left({\text{SU}}_{N_F}\right),$$ where $EG(BG,G)$ is the universal bundle...

Let $X$ be a (generalized) flag manifold of a complex semisimple Lie group $G$. We investigate the problem of constructing a graded star product on $ℛ=R\left(T^{☆}X\right)$ which corresponds to a $G$-equivariant quantization of symbols into twisted differential operators acting on half-forms on $X$. We construct, when $ℛ$ is generated by the momentum functions ${\mu }^x$ for $G$, a preferred choice of $☆$ where ${\mu }^x☆\phi$ has the form ${\mu }^x\phi +\frac{1}{2}\{{\mu }^x,\phi \}t+{\Lambda }^x\left(\phi \right)t^2$. Here ${\Lambda }^x$ are operators on $ℛ$. In the known examples, ${\Lambda }^x$ ($x\ne 0$) is not a differential operator, and so the star product ${\mu }^x☆\phi$...

This paper contains the lectures given by the author at the Winter School on “Geometry and Physics” in Srní 2001. These lectures are based on two recent works of the author with A. Korányi and on a forthcoming paper with K. Johnson and A. Korányi. In the paper results are presented concerning equivariant differential operators on homogeneous spaces (section 1), first order equivariant differential operators on boundaries of symmetric spaces (section 2), the Poisson transform (section 3) and complex...

In this note all vectors and $\epsilon$-vectors of a system of $m\le n$ linearly independent contravariant vectors in the $n$-dimensional pseudo-Euclidean geometry of index one are determined. The problem is resolved by finding the general solution of the functional equation $F(A\underset{1}{\to }u,A\underset{2}{\to }u,\cdots ,A\underset{m}{\to }u)={(detA)}^{\lambda }·A·F(\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{m}{\to }u)$ with $\lambda =0$ and $\lambda =1$, for an arbitrary pseudo-orthogonal matrix $A$ of index one and given vectors $\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{m}{\to }u.$

There are four kinds of scalars in the $n$-dimensional pseudo-Euclidean geometry of index one. In this note, we determine all scalars as concomitants of a system of $m\le n$ linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation $F(A\underset{1}{\to }u,A\underset{2}{\to }u,\cdots ,A\underset{m}{\to }u)=\varphi \left(A\right)·F(\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{m}{\to }u)$ using two homomorphisms $\varphi$ from a group $G$ into the group of real numbers ${ℝ}_0=\left(ℝ\setminus \left\}0\right\{,·\right)$.

In this note, there are determined all biscalars of a system of $s\le n$ linearly independent contravariant vectors in $n$-dimensional pseudo-Euclidean geometry of index one. The problem is resolved by finding a general solution of the functional equation $F(A\underset{1}{\to }u,A\underset{2}{\to }u,\cdots ,A\underset{s}{\to }u)=\left(\text{sign}(detA)\right)F(\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{s}{\to }u)$ for an arbitrary pseudo-orthogonal matrix $A$ of index one and the given vectors $\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{s}{\to }u$.

Se obtiene un nuevo método para obtener toros de Willmore en estructuras conformes de Kaluza-Klein sobre fibrados principales con fibra la circunferencia. Diversas aplicaciones de esta técnica son consideradas.