20 papers · 1 filter
Induced connections on submanifolds in spaces with fundamental groups
Maks A. Akivis, Vladislav V. Goldberg, Arto V. Chakmazyan
The authors establish a relation of the theory of varieties with degenerate Gauss maps in projective spaces with the theory of congruences and pseudocongruences of subspaces and sh…
Dually degenerate varieties and the generalization of a theorem of Griffiths--Harris
Maks A. Akivis, Vladislav V. Goldberg
The dual variety X* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X* is a hypersurface i…
Linearizability of d-webs, d \geq 4, on two-dimensional manifolds
Maks A. Akivis, Vladislav V. Goldberg, Valentin V. Lychagin
We find d - 2 relative differential invariants for a d-web, d \geq 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be lin…
Geometry of Projective Planes over Two-Dimensional Algebras
Maks A. Akivis, Vladislav V. Goldberg
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5…
An affine analogue of the Hartman-Nirenberg cylinder theorem
Maks A. Akivis, Vladislav V. Goldberg
Let X be a smooth, complete, connected submanifold of dimension n < N in a complex affine space A^N (C), and r is the rank of its Gauss map γ, γ(x) = T_x (X). The authors prove tha…
On the Structure of Submanifolds with Degenerate Gauss Mappings
Maks A. Akivis, Vladislav V. Goldberg
An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The auth…