Examples of interesting / non-trivial manifolds that are direct products What are interesting / non-trivial examples of smooth connected closed manifolds that are direct products or involve direct products? I am especially interested in orientable manifolds.
Say, an $n$-torus $T^n$ is a direct product of $n$ copies of a circumference $S^1$. One can build a 3-manifold from a surface of genus $g$ as $M=M^2_g\times S^1$, and use somehow the connected sums of such manifolds.
Typically an "important" manifold would have a name or standard notation. For example, the Kodaira-Thurston manifold (important if it has a proper name!) decomposes into a product of the Heisenberg nil manifold and $S^1$.
I am looking for other important / interesting / non-trivial manifolds that happen to be direct products, preferably having important / interesting applications.

A non degenerate  (=rank $4$) complex quadratic form $q(x,y,z,t)$ defines a smooth   quadric $Q=V(q)=\{q=0\}\subset \mathbb P^3(\mathbb C)$.
This quadric is  a complex manifold of dimension $2$ and thus a compact differential manifold of real dimension $4$.
An elementary but extremely beautiful theorem in algebraic geometry states that every such quadric $Q$ is algebraically isomorphic to $\mathbb P^1(\mathbb C)\times \mathbb P^1(\mathbb C)$ and so a fortiori diffeomorphic to the product $S^2\times S^2$ of two spheres.  
A completely explicit example is given by the quadratic form $q(x,y,z,t)=xt-yz$.
An isomorphism of algebraic varieties between $\mathbb P^1(\mathbb C)\times \mathbb P^1(\mathbb C)$ and the  quadric $Q$ given by the equation  $xt-yz=0$ is supplied by the so-called Segre map:
$$ \mathbb P^1(\mathbb C)\times \mathbb P^1(\mathbb C)\stackrel {\cong}{\to} Q:([t:u],[v:w]) \mapsto  [x:y:z:t]=[tv:tw:uv:uw]                                $$
