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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Reducibility of isometric immersions


Author: John Douglas Moore
Journal: Proc. Amer. Math. Soc. 34 (1972), 229-232
MSC: Primary 53C40
DOI: https://doi.org/10.1090/S0002-9939-1972-0293546-3
MathSciNet review: 0293546
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Abstract: For $ i = 1,2$, suppose that the connected riemannian manifold $ {M_i}$ possesses a codimension $ {p_i}$ euclidean isometric immersion whose first normal space has dimension $ {p_i}$ and whose type number is at least two at each point, and let $ N = \dim ({M_1} \times {M_2}) + {p_1} + {p_2}$. In this note it is proven that if f is any isometric immersion from the riemannian product $ {M_1} \times {M_2}$ into euclidean N-space $ {E^N}$, then there exists an orthogonal decomposition $ {E^N} = {E^{{N_1}}} \times {E^{{N_2}}}$ together with isometric immersions $ {f_i}:{M_i} \to {E^{{N_i}}}$ such that $ f = {f_1} \times {f_2}$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0293546-3
Keywords: Isometric immersion, riemannian product, type number
Article copyright: © Copyright 1972 American Mathematical Society