Analog of Newlander–Nirenberg theorem for real analytic manifolds

It is a theorem of Whitney that every closed $C^{\infty}$-manifold admits a real analytic structure. Furthermore, by a theorem of Morrey and Grauert, this real analytic structure is unique. In any case, as there is no integrability condition to satisfy, I think this probably answers your question in the negative. See this thread Can every manifold be given an analytic structure? for a more detailed discussion.

I think this question can be addressed in a few ways. Two great answers have already been given:

1. Real analytic is a regularity condition, while holomorphic is more algebraic, so you'd need a somewhat different condition than NN.
2. Every $C^1$ manifold can be given a real analytic structure, so you don't even need a condition.

Let me add a third, more complicated, answer where we ask the real analytic structure to be adapted to the given data. To do this we have to go back and look at what the Newlander-Nirenberg theorem gives us, and then we will be able to adapt it to give a real analytic type result. So first

The Newlander-Nirenberg Theorem: Let $L_1,\ldots, L_m$ be smooth complex vector fields on a smooth manifold $M$. Suppose for each $\zeta\in M$ we have the following:

• $[L_j,L_k](\zeta)\in \mathrm{span}_{\mathbb{C}}\{L_1(\zeta),\ldots, L_m(\zeta)\}$.
• $\mathrm{span}_{\mathbb{C}} \{ L_1(\zeta),\ldots, L_m(\zeta)\} \cap \mathrm{span}_{\mathbb{C}} \{ \overline{L_1}(\zeta),\ldots, \overline{L_m}(\zeta)\}=\{0\}$.
• $\mathrm{span}_{\mathbb{C}}\{L_1(\zeta),\ldots, L_m(\zeta), \overline{L_1}(\zeta),\ldots, \overline{L_m}(\zeta)\} = \mathbb{C} T_{\zeta} M$.

Then, there exists a complex manifold structure on $M$ (compatible with its smooth manifold structure) such that $\forall \zeta\in M$, $\mathrm{span}_{\mathbb{C}} \{L_1(\zeta),\ldots, L_m(\zeta)\} = T^{0,1}M$. Moreover, this complex structure is unique in the sense that if $M$ is given another complex manifold structure (compatible with its smooth manifold structure) such that $\mathrm{span}_{\mathbb{C}} \{L_1(\zeta),\ldots, L_m(\zeta)\} = T^{0,1}M$, then the identity map $M\rightarrow M$ is a biholomorphism between these two complex manifold structures on $M$.

Now let's turn to the real setting, and address the original question. Let $X_1,\ldots, X_q$ be $C^1$ vector fields on a $C^2$ manifold $M$ such that $\forall x\in M$, $\mathrm{span}_{\mathbb{R}} \{X_1(x),\ldots, X_q(x)\} = T_x M$.

Question: When is there a real analytic structure on $M$, compatible with its $C^2$ structure, such that $X_1,\ldots, X_q$ are real analytic with respect to this structure? When such a structure exists, we will see it is unique (in the strongest possible sense).

We will answer this question by giving a condition which looks very similar to the Newlander-Nirenberg theorem (but is actually orthogonal to that condition, as we will see).

Even though $M$ is only a $C^2$ manifold, we can define what it means for a function to be real analytic with respect to the vector fields $X_1,\ldots, X_q$. Let $V\subseteq M$ be open. For $r>0$ we define $C_{X}^{\omega,r}(V)$ to be the space of those $f:V\rightarrow \mathbb{R}$ such that the following norm is finite: $$ \| f\|_{C_X^{\omega,r}(V)} = \sum_{m=0}^{\infty} \frac{r^m}{m!} \sum_{|\alpha|=m} \| X^{\alpha} f\|_{C(V)}.$$ In defining $X^{\alpha}$ we have used ordered multi-index notation; i.e., $\alpha$ is a list of elements of $\{1,\ldots, q\}$ and $|\alpha|$ is the length of the list. For example $X^{(1,2,1,2)}= X_1X_2X_1X_2$ and $|(1,2,1,2)|=4$. We set $C_X^{\omega}(V):= \bigcup_{r>0} C_X^{\omega,r}(V)$. The space $C_X^{\omega}(V)$ is "coordinate-free": it does not depend on any choice of coordinate system or atlas. This space was originally defined in greater generality by Nelson (Ann. of Math. (2) 70 (1959), 572-615).

Theorem: There is a real analytic structure on $M$, compatible with its $C^2$ structure, such that $X_1,\ldots, X_q$ are real analytic with respect to this structure if and only if:

• For every $x\in M$, there is a neighborhood $V_x$ of $x$, and functions $c_{j,k}^{l,x}\in C_{X}^{\omega}(V_x)$ such that $[X_j,X_k]=\sum_{l=1}^q c_{j,k}^{l,x} X_l$ on $V_x$.

When this real analytic structure exists, it is unique in the sense that if $M$ is given another real analytic structure, compatible with its $C^2$ structure, such that $X_1,\ldots, X_q$ are real analytic with respect to this second structure, then the identity map $M\rightarrow M$ is a real analytic diffeomorphism between these two real analytic structures.

So there's the answer to our question--and it has the same "feel" as the NN theorem: it uses an understanding of commutators of given vector fields to give a unique structure on the manifold.

There's nothing special about real analytic in the above. You can replace real analytic with an appropriate space of functions which are $C^\infty$ with respect to $X_1,\ldots, X_q$ and get a corresponding result about $C^\infty$ structures. You can do it for a finite level of smoothness, too, though to obtain a sharp if and only if Zygmund spaces are used (instead of the more familiar $C^m$ spaces).

This is all done in the series of papers (joint with Stovall): 1, 2, 3

No back to my comment that this was actually orthogonal to the NN theroem. One could ask the following:

Question: Suppose $L_1,\ldots, L_m$ satisfy the conditions of the NN theorem (above). When are $L_1,\ldots, L_m$ real analytic with respect to the complex structure gauranteed by the NN theorem?

One can answer this by giving a condition very similar to the real theory given above. You can even get a theory which unifies both the real and complex into one theorem. This is all contained in this paper.