### A quaternionic treatment of Navier-Stokes equations

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[For the entire collection see Zbl 0699.00032.] The author defines a general notion of a foliated groupoid over a foliation with singularities, within the framework of a (known) general notion of a differentiable structure. Then, he generalizes the classical correspondence between the subalgebras of Lie algebras and the subgroups of the corresponding Lie groups for this type of pseudogroups.

[For the entire collection see Zbl 0699.00032.] A manifold (M,g) is said to be generalized Einstein manifold if the following condition is satisfied $$\left({\nabla}_{X}S\right)(Y,Z)=\sigma \left(X\right)g(Y,Z)+\nu \left(Y\right)g(X,Z)+\nu \left(Z\right)g(X,Y)$$ where S(X,Y) is the Ricci tensor of (M,g) and $\sigma $ (X), $\nu $ (X) are certain $\ell $-forms. In the present paper the author studies properties of conformal and geodesic mappings of generalized Einstein manifolds. He gives the local classification of generalized Einstein manifolds when g($\psi $ (X),$\psi $ (X))$\ne 0$.

[For the entire collection see Zbl 0699.00032.] The paper deals with a special problem of gauge theory. In his previous paper [The invariance of Sobolev spaces over noncompact manifolds, Partial differential equations, Proc. Symp., Holzhaus/GDR 1988, Teubner- Texte Math. 112, 73-107 (1989; Zbl 0681.58011)], the author introduced the Sobolev completions ${\overline{\mathcal{C}}}_{P}^{k}$ of the space ${\mathcal{C}}_{P}$ of all G-connections on a G-principal fibre bundle P. In the present paper, under the assumption of bounded curvatures and their...

Let $A={\u2a01}_{k}{A}_{k}$ and $B={\u2a01}_{k}{B}_{k}$ be graded Lie algebras whose grading is in $\mathcal{Z}$ or ${\mathcal{Z}}_{2}$, but only one of them. Suppose that $(\alpha ,\beta )$ is a derivatively knitted pair of representations for $(A,B)$, i.e. $\alpha $ and $\beta $ satisfy equations which look “derivatively knitted"; then $A\oplus B:={\u2a01}_{k,l}({A}_{k}\oplus {B}_{l})$, endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra $A{\oplus}_{(\alpha ,\beta )}B$. This graded Lie algebra is called the knit product of $A$ and $B$. The author investigates the general situation for any graded Lie subalgebras $A$ and $B$ of a graded...

[For the entire collection see Zbl 0699.00032.] Natural transformations of the Weil functor ${T}^{A}$ of A-velocities [I. Kolař, Commentat. Math. Univ. Carol. 27, 723-729 (1986; Zbl 0603.58001)] into an arbitrary bundle functor F are characterized. In the case where F is a linear bundle functor, the author deduces that the dimension of the vector space of all natural transformations of ${T}^{A}$ into F is finite and is less than or equal to $dim\left({F}_{0}{\mathcal{R}}^{k}\right)$. The spaces of all natural transformations of Weil functors into linear...

[For the entire collection see Zbl 0699.00032.] A connection structure (M,H) and a path structure (M,S) on the manifold M are called compatible, if $S\left(v\right)=H(v,v),\forall v\in TM,$ locally ${G}^{i}(x,y)={y}^{j}{\Gamma}_{j}^{i}(x,y),$ where ${G}^{i}$ and ${\Gamma}_{j}^{i}$ express the semi-spray S and the connection map H, resp. In the linear case of H its geodesic spray S is quadratic: ${G}^{i}(x,y)={\Gamma}_{jk}^{i}\left(k\right){y}^{j}{y}^{k}.$ On the contrary, the homogeneity condition of S induces the relation for the compatible connection H, ${y}^{j}({\Gamma}_{j}^{i}\circ {\mu}_{t})=t{y}^{j}{\Gamma}_{j}^{i},$ whence it follows not that H is linear, i.e. if a connection structure is compatible with a spray, then...

[For the entire collection see Zbl 0699.00032.] The author considers the conformal relation between twistors and spinors on a Riemannian spin manifold of dimension $n\ge 3$. A first integral is constructed for a twistor spinor and various geometric properties of the spin manifold are deduced. The notions of a conformal deformation and a Killing spinor are considered and such a deformation of a twistor spinor into a Killing spinor and conditions for the equivalence of these quantities is indicated.

[For the entire collection see Zbl 0699.00032.] A new cohomology theory suitable for understanding of nonlinear partial differential equations is presented. This paper is a continuation of the following paper of the author [Differ. geometry and its appl., Proc. Conf., Brno/Czech. 1986, Commun., 235-244 (1987; Zbl 0629.58033)].

[For the entire collection see Zbl 0699.00032.] It was previously known that for every principal fibre bundle P there is some corresponding transitive Lie algebroid A(P) - a vector bundle equipped with some structure like the structure of a Lie algebra in the module of sections. The author of this article shows that the Chern-Weil homomorphism of P is a notion of the Lie algebroid of P, i.e. knowing only A(P) of P one can uniquely reproduce the ring of invariant polynomials ${\left(V{g}^{*}\right)}_{I}$ and the Chern-Weil...