Janet theorem

In every analytic Riemannian manifold of dimension $ n $ there exists a neighbourhood of an arbitrarily chosen point having an isometric imbedding into the Euclidean space $ \mathbf R ^ {s _ {n} } $ of dimension $ s _ {n} = n ( n + 1 ) / 2 $. Janet's theorem remains true if $ \mathbf R ^ {s _ {n} } $ is replaced by any analytic Riemannian manifold of dimension $ s _ {n} $ with a prescribed point (to which the point chosen in the original manifold must be mapped). Janet's theorem is valid in the case of pseudo-Riemannian manifolds provided that

$$ q \geq s _ {n} ,\ q _ {+} \geq n _ {+} ,\ \ q _ {-} \geq n _ {-} , $$

where $ n _ {+} $ and $ n _ {-} = n - n _ {+} $ are the dimensions of the positive and negative parts of the metric tensor on the original manifold, and $ q _ {+} $ and $ q _ {-} = q - q _ {+} $ are the corresponding dimensions of the target manifold (see [3]). Janet's theorem is the first general imbedding theorem in Riemannian geometry (see Isometric immersion).

Janet's theorem first appeared as a conjecture of L. Schläfli [1], and was proved by M. Janet [2].

References

[1]	L. Schläfli, "Nota alla Memoria del signor Beltrami "Sugli spazi di curvatura costante" " Ann. Mat. Pura. Appl. Ser. 2 , 5 (1873) pp. 178–193
[2]	M. Janet, "Sur la possibilité de plonger un espace Riemannien donné dans un espace euclidien" Ann. Soc. Polon. Math. , 5 (1926) pp. 38–43
[3]	A. Friedman, "Isometric imbedding of Riemannian manifolds into Euclidean spaces" Rev. Modern Physics , 77 (1965) pp. 201–203

Comments

The theorem was also proved by E. Cartan [a1]. A rigorous proof along the lines suggested by Janet was given by C. Burstin [a2]. See also [a3].

References

[a1]	E. Cartan, "Sur la possibilité de plonger un espace riemannien donné dans un espace euclidéen" Ann. Soc. Polon. Math. , 6 (1927) pp. 1–7
[a2]	C. Burstin, Mat. Sb. , 38 (1931) pp. 74–93
[a3]	M. Spivak, "A comprehensive introduction to differential geometry" , 5 , Publish or Perish (1975) pp. 1–5