# Fundamental theorem for real submanifolds into complex space forms

It is a well-known result that Gauss, Codazzi and Ricci equations are necessary and suficiently conditions to guarantee the existence of an isometric immersion of a given $$n$$-dimensional real Riemannian manifold into a $$(n+p)$$-dimensional real space form.

I would like to know if there is some version of this result for complex space forms. Can someone give me a reference?

• Yes. You can find the proof here: Griffiths, P. On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. J. 41 (1974), 775–814. Dec 10, 2020 at 19:33
• Thanks for your comments @DeaneYang But, are you sure that the proof for real submanifolds into complex space forms is contained there? Because, in this article I found only the proof in Euclidean n-space. Can you point out the pages, for example? Dec 11, 2020 at 3:11
• I concede that there's no explicit statement of this result, but it is a special case of the two lemmas (1.3) and (1.4). Very roughly speaking, it goes like this: Let $G$ be the group of isometries of the complex space form and $\mathfrak{g}$ its Lie algebra. The assumption that the Gauss, Codazzi, and Ricci equations holds is equivalent to there being a $\mathfrak{g}$-valued $1$-form on the real manifold that satisfies the Maurer-Cartan equation. Lemmas (1.3) and (1.4) imply the existence of the isometric immersion. Dec 11, 2020 at 5:06
• In particular, the proof in section 3 can be adapted to the case where the ambient space is a complex space form instead of Euclidean space. Dec 11, 2020 at 5:13
• @DeaneYang: I agree with your comments, but I think that the real issue is what the givens are in the complex space form theorem. In the real case, the givens are a manifold $M^n$, a metric $g$ on $M$, an Euclidean vector bundle $N$ of rank $p$ over $M$ endowed with an Euclidean connection, and a section $I\!I$ of $N\otimes S^2(T^*M)$, all assumed to satisfy Gauss, Codazzi, and Ricci. In the complex case, one also needs an orthogonal complex structure $J$ on $TM\oplus N$ plus some equations that relate it to the other data. It is only at this point that Cartan's Fundamental Lemma takes over. Dec 15, 2020 at 11:58