Beitraege zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 34 (1993), No. 2, 209-215.}
The Semiring of Immersions of Manifolds
F. Decruyenaere, F. Dillen, L. Verstraelen, L. Vrancken
Abstract.
In [2], [3], [4], [5] B.-Y.\ Chen introduced the tensor product
immersion of a given Riemannian manifold. In this paper we study the
tensor product of two immersions of differentiable manifolds.
Although inspired by Chen's definition, our tensor product is a
somewhat different concept, since it realizes an immersion of the
product manifold. Basically, we will obtain a commutative semiring
structure on the set of transversal immersions of manifolds in
Euclidean spaces with operations direct sum $\oplus$ and tensor
product $\otimes$. As a geometrical application of our notion of
tensor products, we will make a remark on a conjecture of H.\ Hopf
concerning Riemannian metrics on $S^2\times S^2$.
1991 Mathematics Subject Classification. 53C40, 53B25, 58G25