Bibliographical references to Hilbert's theorem

Question

I want to do a thorough investigation of Hilbert's Theorem and them Efimov's Theorem:

(Hilbert's Theorem) A complete surface $S$ with constant negative curvature can't be isometrically immersed in $R^3$.

(Efimov's Theorem) A complete surface with Gauss curvature $K\leq k_0<0$ can't be isometrically immersed in $R^3$.

and I would like to know if anyone knows good bibliographical references on it apart from the book by Do Carmo. I would really appreciate it.

Answer 1

Here are some papers/links on imbedding surfaces of negative curvature:

• Mathoverflow post: Surfaces in $\mathbb R^3$ with negative curvature bounded away from zero

• Efimov, N. V., Impossibility of isometric immersion into three-dimensional Euclidean space of certain manifolds with negative Gaussian curvature, Sov. Math., Dokl. 3 (1962), 1293-1297 (1963); translation from Dokl. Akad. Nauk SSSR 146, 296-299 (1962). ZBL0137.40702.

• Efimov, N. V., Impossibility of a complete regular surface in Euclidean 3-space whose Gaussian curvature has a negative upper bound, Sov. Math., Dokl. 4, 843-846 (1963); translation from Dokl. Akad. Nauk SSSR 150, 1206-1209 (1963). ZBL0135.40001.

• Han, Qing; Hong, Jia-Xing, Isometric embedding of Riemannian manifolds in Euclidean spaces, Mathematical Surveys and Monographs 130. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4071-1/hbk). xiv, 260 p. (2006). ZBL1113.53002.

• Klotz Milnor, Tilla, Efimov’s theorem about complete immersed surfaces of negative curvature, Adv. Math. 8, 474-543 (1972). ZBL0236.53055.

• J J Stoker, ON THE EMBEDDING OF SURFACES OF NEGATIVE CURVATURE IN 3-DIMENSIONAL EUCLIDEAN SPACE

• Shikin, E. V., Equations of isometric imbeddings in three-dimensional Euclidean space of two-dimensional manifolds of negative curvature, Math. Notes 31, 305-312 (1982). ZBL0499.53004.

• Poznyak, É.G., Shikin, E.V. Surfaces of negative curvature. J Math Sci 5, 865–887 (1976).