**Summary of comments and other sources**

There are at least 4 similar concepts:

**Irreducible smooth manifold:** As Ryan Budney said, "Regarding high dimensions, generally irreducible manifolds do not exist, this is because the connect-sum operation has some invertible objects -- in dimension 5 and up they are known as homotopy-spheres." See more in

*Kosinski, Antoni A.*, Differential manifolds, Pure and Applied Mathematics, 138. Boston, MA: Academic Press. xvi, 248 p. (1993). ZBL0767.57001.

and

*Kervaire, Michel A.; Milnor, John W.*, **Groups of homotopy spheres. I**, Ann. Math. (2) 77, 504-537 (1963). ZBL0115.40505.

**Irreducible symmetric spaces:** A symmetric space is
said to be irreducible if it is not isometric to a product of positive-dimensional
symmetric spaces.

**Irreducible vector space:** A vector space is said to be irreducible if has no nontrivial invariant subspaces inside it.

and using case 3 we can talk about

**Irreducible Riemannian manifolds:** Those manifolds where the holonomy has no invariant subspaces.

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