Local smooth isometric embeddings of low dimensional riemannian manifolds into euclidean spaces

Gen Nakamura, Yoshiaki Maeda

    Research output: Contribution to journalArticlepeer-review

    20 Citations (Scopus)

    Abstract

    Local smooth isometric embedding problems of low dimensional Riemannian manifolds into Euclidean spaces are studied. Namely, we prove the existence of a local smooth isometric embedding of a smooth Riemannian 3-manifold with nonvanishing curvature into Euclidean 6-space. For proving this, we give a local solvability theorem for a system of a nonlinear PDE of real principal type. To obtain the local solvability theorem, we need a tame estimate for the linearized equation corresponding to the given PDE, which is presented by two methods. The first is based on the result of Duistermaat-Hörmander which constructed the exact right inverse for linear PDEs of real principal type by using Fourier integral operators. The second method uses more various properties of Fourier integral operators given by Kumano-go, which seems to be a simpler proof than the above.

    Original languageEnglish
    Pages (from-to)1-51
    Number of pages51
    JournalTransactions of the American Mathematical Society
    Volume313
    Issue number1
    DOIs
    Publication statusPublished - 1989 May

    ASJC Scopus subject areas

    • Mathematics(all)
    • Applied Mathematics

    Fingerprint Dive into the research topics of 'Local smooth isometric embeddings of low dimensional riemannian manifolds into euclidean spaces'. Together they form a unique fingerprint.

    Cite this