Skew embeddings and immersions of manifolds into Euclidean spaces in codimension two

by Tyurina, Yulia.

Embeddings and immersions of smooth closed oriented manifolds into Euclidean spaces of codimension two are investigated. An embedding (immersion) is skew if the embedded (immersed) manifold does not have a pair of parallel tangent spaces. Each tangent space inherits the orientation of the manifold. Parallel tangent spaces with the same orientation are called positively parallel, with opposite orientation - negatively parallel. The author shows that an even-dimensional manifold cannot be skew embedded if its Euler characteristic is not zero, and provides an example of a skew embedding of a two-dimensional torus into R4. The author also proves that a skew embedding exist for a sphere of an arbitrary odd dimension. For immersions of two-dimensional oriented closed manifolds into R4, the author gives an estimate from below for the number of negatively parallel tangents. The author constructs an example of an immersion of S2 into R4 with one point of self-intersection, for which this estimate is sharp. iii