A Hermitian symmetric space is a connected complex manifold with a hermitian metric on which the group of holomorphic isometries acts transitively, and which satisfies the following extra condition: each point (equivalently, some point) is an isolated fixed point of some (holomorphic, isometric) involution of the space. There is a beautiful classification theory of Hermitian symmetric spaces, and this is the starting point for the study of Shimura varieties.


The first four conditions (connected, complex, hermitian, homogeneous), which define a Hermitian homogeneous space, all seem natural enough to me. My question: why the need for the extra condition regarding the involution?


This is deliberately vague because I'm not quite sure what form the most satisfying answer will take. But here are a couple of more specific versions of the question. (1) What sort of spaces are we excluding by imposing this condition, and why? (2) Are there other conditions which might seem more natural to me which are equivalent (or equivalent some of the time)? For instance if you were to tell me that a Hermitian homogeneous space that's negatively curved and simply-connected always has such an involution, I would be completely satisfied.



Hermitian symmetric spaces vs Hermitian homogeneous spaces
Here is a geometric answer to (2), or more precisely a slight and classical reinterpretation. First, note that Hermitian symmetric spaces fit into the more general concept of riemannian symmetric space; this can helps you find references.


Denote your space by $X$ (assumed to be Riemannian, or Hermitian if you prefer) and take a point $p$. Consider the geodesic symmetry around $p$: it maps a point $q$ to the point $\sigma\_p(q)$ so that $\sigma\_p(q), p, q$ lie on a constant speed geodesic, at times $-1,0,1$ respectively. This map is at least defined locally. Then $X$ is symmetric if and only if for all $p\in X$, $\sigma\_p$ is globally defined and an isometry (ask it to be also holomorphic if your are in the Hermitian setting, but I guess it will automatically be so since it is conjugate to $-\mathrm{Id}$ by the exponential map). Of course, $\sigma\_p$ is your involution.


This condition automatically implies that the isometry group of $X$ acts transitively (because any $q$ is mapped to any $q'$ by the map $\sigma\_p$ where $p$ is a midpoint of $[q,q']$).


It also gives you an involution $\theta$ on the Lie algebra $\mathfrak{g}$ of the isometry group of $X$. Now remark that $\theta$ is a linear endomorphism, and $\theta^2=1$ implies that
$\mathfrak{g}$ decomposes into two components, the eigenspace associated to the eigenvalue $1$ of $\theta$ and the one associated to $-1$. They are usually denoted by $\mathfrak{h}$ and $\mathfrak{p}$; the latter identifies with the tangent space to $X$ at $p$. This is called Cartan decomposition and is fundamental to the study of these spaces.


The most common reference is Helgason's *Differential geometry, Lie groups, and symmetric spaces* but I find it quite difficult to read. In the nonpositively curved case, I found Eberlein's *Geometry of nonpositively curved manifolds* useful.


Concerning the motivation, it seems that the interest in symmetric spaces is that they provide examples between constant curvature spaces, which are very constrained, and homogeneous spaces that are very numerous.


Last, a lead concerning (1). You shall find in Berger's *A panoramic view of Riemannian geometry*, section 15.8.1 page 719 a formula enabling one to compute the curvature of a homogeneous space. I guess it can be used to show the existence of many such spaces, even in the Hermitian world, that are negatively curved but not symmetric.


