Global rigidity of compact classical Lie groups
Let G(n,F) denote the compact classical Lie group SO(n), U(n) or Sp(n) depending on \(F={\mathbb{R}}\), \({\mathbb{C}}\) or \({\mathbb{Q}}\), the field of quaternions. In this article the author deals with the problem of rigidity of G(n,F) imbedded isometrically in the space of \(n\times n\) matrices over F. We recall that an imbedding \(f: M\to {\mathbb{R}}^ m\), M a Riemannian manifold, is elliptic if i) the second fundamental form \(\alpha\) (x,y) generates the normal space at \(p\in M\), for all p; ii) for every normal direction \(\xi\), \(A_{\xi}\) has at least two non zero eigenvalues of the same sign \((<A_{\xi}X,Y>=<\alpha (x,y),\xi >)\). In Nagoya Math. J. 51, 137-160 (1973; Zbl 0265.53050), \textit{N. Tanaka} introduced the notion of elliptic imbedding and proved a rigidity theorem which may be stated as follows: Let f be an imbedding such that a) f is elliptic, b) it is infinitesimally rigid, c) M is compact. If \(f_ 1\) and \(f_ 2\) are two imbeddings of M in \({\mathbb{R}}^ m\) which lie near f with respect to the \(C^ 3\) topology and induce the same metric on M then there is a unique Euclidean transformation T of \({\mathbb{R}}^ m\) such that \(f_ 2=Tf_ 1.\) In the present paper it is proved that the inclusion map of SO(n), \(n\geq 5\), U(n), \(n\geq 3\), and Sp(n), \(n\geq i\) is elliptic and globally infinitesimally rigid hence rigid in the sense of Tanaka.
scientific article; zbMATH DE number 3947124