# 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

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
How to Cite This Entry:
Janet theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Janet_theorem&oldid=47462
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article