Grauert's approximation theorem
In mathematics, Grauert's approximation theorem, due to Grauert, is an analog of Whitney’s approximation theorem for real-analytic maps. It states: with respect to the Whitney topology (also known as strong topology), the space of real-analytic maps between real-analytic manifolds is dense in the space of smooth maps between those manifolds.[1] In the compact case, the theorem is due to Morrey.[2] The case when there is an analytic Riemannian metric is due to Bochner.[3]
See also
References
- ^ Hirsch 1976, Ch. 2., § 5., Theorem 5.1.
- ^ Morrey, Charles B. (1958). "The Analytic Embedding of Abstract Real-Analytic Manifolds". Annals of Mathematics. 68 (1): 159–201. doi:10.2307/1970048. ISSN 0003-486X.
- ^ Bochner, S. (1937). "Analytic mapping of compact Riemann spaces into Euclidean space". Duke Math. J. (in French). 3 (1): 339–354.
- Grauert, Hans (1958). "On Levi's Problem and the Imbedding of Real-Analytic Manifolds". Annals of Mathematics. 68 (2): 460–472. doi:10.2307/1970257. ISSN 0003-486X.
- Hirsch, Morris W. (1976). Differential topology. New York Heidelberg Berlin: Springer-Verlag. ISBN 978-1-4684-9449-5.