Flatness as an integrability condition in the context of Riemannian geometry

Question

I am reading Y. Eliashberg and N. Mishachev's book Introduction to the $h$-principle. At the beginning of section 9.2.F, on page 82, the authors say:

Note that a symplectic (complex) submanifold $S\subset V$ of a symplectic (complex) manifold $V$ ingerits an integrable symplectic (complex) structure. This contrasts with the Riemannian case: the symplectic integrability condition is analogous to the condition in the Riemannian case of being locally Euclidean, but submanifolds of locally Euclidean manifolds need not, of course, be locally Euclidean.

In the symplectic setting, integrable means that the nondegenerate 2-form $\omega$ on $V$ provided by an almost symplectic structure is indeed a symplectic structure, i.e., it satisfies the condition $d\omega=0$. On the other hand, integrable in the complex case means that the almost complex structure $J$ on $V$ satisfies that its Nijenhuis tensor vanishes.

I am terribly confused about the aforementioned statement; I actually don't know what the authors are trying to express. For instance, what do we mean by locally Euclidean? All manifolds are locally Euclidean. Maybe the authors are pointing out that, in symplectic geometry, all symplectic manifolds are built up from Darboux coordinates glued toghether by symplectomorphisms, and that Riemannian manifolds, in contrast, are not generally flat, or rather, flatness is not a property that is preserved by taking the restriction of the metric to the corresponding submanifold.

But again, why being flat should be considered an integrability condition? It makes sense from the point of view of its construction, that is, flat manifolds are built from charts to Euclidean space with the usual metric, glued toghether by isometries, but I think I am missing to what this structure integrates to.

Thanks in advance for your answers.

Answer 1

You can consider three different situations:

1. An almost symplectic manifold $(M,\omega)$.

2. An almost complex manifold $(M,J)$.

3. A Riemannian manifold $(M,g)$.

In all these three cases it makes sense to ask the question of the existence of local diffeomorphisms $\varphi:\mathbb{R}^n \rightarrow U\subset M$ such that they respect, respectively, the almost symplectic structure $\omega$, the almost complex structure $J$, or the Riemannian structure $g$. This means that, respectively, $\varphi^* \omega$, $\varphi^* J$ or $\varphi^* g$ are the standard symplectic structure, complex structure, and Riemannian structure in the Euclidean space.

If you write down $\varphi$ in coordinates (that is, $(x_1\dots,x_n)\mapsto (y_1,\dots,y_n)$), you get that all these three conditions can be expressed as PDEs for the $y_i(x_1,\dots,x_n)$. These PDEs admit solutions if and only if some integrability conditions are satisfied. These integrability conditions are in each case:

1. $d\omega=0$ (this is the Darboux theorem)
2. $N_J=0$, where $N_J$ is the Nijenhuis tensor (this is the Newlander--Niremberg theorem),
3. $R=0$, where $R$ is the Riemann curvature tensor, that is, the curvature of the Levi-Civita connection, (this is --a consequence of-- the Frobenius theorem).

By the way, notice that 2 is a much more difficult result than 1 and 3.