Janet theorem
In every analytic Riemannian manifold of dimension there exists a neighbourhood of an arbitrarily chosen point having an isometric imbedding into the Euclidean space
of dimension
. Janet's theorem remains true if
is replaced by any analytic Riemannian manifold of dimension
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
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where and
are the dimensions of the positive and negative parts of the metric tensor on the original manifold, and
and
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 |
Janet theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Janet_theorem&oldid=47462