Let M be a compact differentiable manifold of class Cr, 1 <r < 00, and let Cr(M) be the space of all real functions of class Cr on M. In addition to the obvious algebraic structure of Cr(M), we shall use a normed algebra structure, the norm being obtained by introducing a Riemannian metric on M. The main results (Theorems 1 and 3) will be stated and proved in this section, using lemmas on differentiable manifolds which will be proved in the following s.ection. Theorem 1 is straightforward, Theorem 3 is more difficult.
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S. B. Myers (1954) studied this question.