Geometry of tensor product immersions

Tensor product map of two immersions from different manifolds into Euclideanspaces is an immersion of the product manifold if and only if at least one of theimmersions is transversal and both the images of the immersions do not contain theorigins of the Euclidean spaces. The general situation and some special situationsfor such a tensor product immersion are considered by various authors.In this work, we first give a brief description of the general situation. Then, weconsider the tensor product of a surface in Euclidean space and a curve in Euclideanplane and we classify minimal, pseudo-umbilical, invariant and anti-invariant suchtensor products. We also give a result which states a necessary and sufficientcondition for one of the factor of the product manifold to be totally geodesicsubmanifold of this product manifold.In the next section, we consider the tensor products of a surface in Euclideanspace and a nondegenerate curve (with a nondegenerate position vector) in Lorentzianplane. We classify minimal, invariant and anti-invariant such tensor products.We also consider the problem from the viewpoint of affine differential geometryand obtain the following results. Tensor product immersion of two centroaffinehypersurface immersions is an equi-centroaffine immersion if and only if these twoimmersions are both nondegenerate. If tensor product of two affine planar curves isan immersion, then this immersion gives a purely real 2-surface in R^4.