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Classification of tensor product immersions which are of 1-type

Published online by Cambridge University Press:  18 May 2009

Bang-Yen Chen
Bang-Yen Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027, U.S.A.
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Let V and W be two vector spaces over the field of real numbers R. Then we have the notion of the tensor product V ⊗ W. If V and W are inner product spaces with their inner products given respectively by «,»v and «,» w, then V ⊗ W is also an inner product space with inner product denned by

Let Em denote the m-dimensional Euclidean space with the canonical Euclidean inner product. Then, with respect to the inner product defined above, Em ⊗Em is isometric to Em. By applying this algebraic notion, we have the notion of tensor product mapf ⊗h: M→ E: M ⊗= Em; associated with any two maps f: M→Em and h:M→E of a given Riemannian manifold (M, g) defined as follows:

Denote by R(M) the set of all transversal immersions from an n-dimensional Riemannian manifold (M, g) into Euclidean spaces; i. e., immersions f:M→Em with f(p) ∉T*(TPM) for p ∈ M. Then ⊗ is a binary operation on R(M). Hence, if f: Mm and h: M→Em are immersions belonging to R(M), then their tensor product map f ⊗ h: M→ Em ⊗ Em ≡ Emm is an immersion in R(M), called the tensor product immersionof f and h.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 2 , May 1994 , pp. 255 - 264
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

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