Hermitian symmetric space

Remarks

Conditions 2 and 3 force $(ds_p)_p = -\id$, so $(M, g)$ is a Riemannian symmetric space and $s_p$ is its geodesic symmetry. The metric $g$ is then Kähler.

Example: the unit disc $D = \set{z \in \C : \abs{z} < 1}$ with the metric $\frac{4\,\abs{dz}^2}{(1 - \abs{z}^2)^2}$. At $0$ take $s_0(z) = -z$: it is holomorphic, an isometry, an involution, and $0$ is its only fixed point. The holomorphic isometries $z \mapsto e^{i t} \frac{z - a}{1 - \bar a z}$ act transitively, and conjugating $s_0$ by them gives $s_p$ at every point $p$.

Non-example: the round sphere $S^4$ is a Riemannian symmetric space, but it has no almost complex structure at all, so it is not a Hermitian symmetric space.

Each simply connected Hermitian symmetric space is a product of a flat part $\C^k$ and irreducible ones of compact type, such as $\C P^n$, and of noncompact type, such as $D$.