If $V$ is a real vector space, then the **complexification** of $V$ is formally defined as $V^{\mathbb{C}}=V\otimes\_{\mathbb{R}}\mathbb{C}$. Is there an analogous complexification operation for a real $n$-dimensional Riemannian manifold $(M,g)$? 


*Idea:* The notion of complexification exists for Lie groups, so perhaps one can "complexify" a real Riemannian manifold by realizing it as a Lie group (or the quotient of one). It seems that under complexification of a real manifold some additional information must be added to determine a complex structure.




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The reason I ask this is because I am looking through the Riemannian holonomy section of [this article](https://en.wikipedia.org/wiki/Holonomy#Riemannian_holonomy) and it states that "the complexified holonomies $SO(n,\mathbb{C})$, $G\_2(\mathbb{C})$, and $Spin(7,\mathbb{C})$ may be realized from complexifying real analytic Riemannian manifolds." What precisely does complexifying a real analytic Riemannian manifold mean in this context?


Any help would be much appreciated!



How does one complexify a real $n$-dimensional Riemannian manifold $(M,g)$?
I believe the following is meant:


Every smooth (real) manifold $M$ has a (unique) real-analytic structure compatible with the smooth structure. So, cover $M$ with real-analytic charts, i.e. whose transition functions are real-analytic diffeomorphisms
$$
\phi\_{ij}:=\phi\_j^{-1}\circ\phi\_i: U\_{ij}:=\phi\_i^{-1}(\phi\_i(U\_i)\cap\phi\_j(U\_j))\to U\_{ji}
$$
One can find open subsets $U\_i^{\mathbb{C}}\subseteq\mathbb{C}^n$ with $U\_i^{\mathbb{C}}\cap\mathbb{R}^n=U\_i$ and $U\_{ij}^{\mathbb{C}}\cap\mathbb{R}^n=U\_{ij}$ such that the (real-analytic) $\phi\_{ij}$ extend to biholomorphisms $\phi\_{ij}^{\mathbb{C}}:U\_{ij}^{\mathbb{C}}\to U\_{ji}^{\mathbb{C}}$ satisfying the usual cocycle conditions. Then the complexification $M^{\mathbb{C}}$ is defined as a quotient space of the disjoint union, $\left(\coprod\_i U\_i^{\mathbb{C}}\right)/\sim$, where $z\_i\sim z\_j$ iff $z\_i\in U\_{ij}^{\mathbb{C}}$ and $z\_j = \phi\_{ij}^{\mathbb{C}}(z\_i)$ (this works because of the cocycle conditions). The maps $U\_i^{\mathbb{C}}\hookrightarrow\coprod U\_i^{\mathbb{C}}$ induce coordinate charts $U\_i^{\mathbb{C}}\to M^{\mathbb{C}}$ with biholomorphic transition functions.


This and the details around it are part (of the proof of) Bruhat-Whitney's theorem\* on the existence of $M^{\mathbb{C}}$. Moreover, complexification is functorial in the obvious way. By Grauert, $M^{\mathbb{C}}$ is in fact a Stein manifold.


\*F. Bruhat and H. Whitney, Quelques propriétés fondamentales des ensembles analytiques-réels, Comment. Math. Helv. 33, 132-160 (1959).


