When does $\mathrm{Diff}(M)$ equal its connected component of the idendity?

Let $M^n$ be a Riemannian compact manifold. What are the weakest conditions that $M$ should satisfy, in order for the connected component of the identity in $\mathrm{Diff}(M)$, $\mathrm{Diff}^0(M)$, be all $\mathrm{Diff}(M)$?

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You mean the connected component of the identity? –  Qiaochu Yuan Dec 1 '12 at 22:56
Rather than looking for the weakest condition, could you say any conditions at all, other than restating that every diffeomorphism is isotopic to the identity? –  Ryan Budney Dec 1 '12 at 23:00
@QiaochuYuan, yes thanks. –  Jorge Campos Dec 1 '12 at 23:14
Most manifolds will not satisfy this property. Any invariant under isotopy, say the action on homology or $\pi_1$, gives a partition into different connected components. In fact, apart from just the manifold that is a single point and $\mathbb{R}P^2$ I can't think of any examples this quickly. –  skupers Dec 2 '12 at 0:46