I would like to know examples of smooth compact connected manifolds, on which there exists an effective smooth circle action preserving a positive smooth volume, besides the simple example: $[0,1]^d \times \mathbb{S}^1$. Is there a classification of these manifolds?



Examples of manifolds with effective circle actions?
As already pointed out, whenever you have a compact group $G$ acting on a manifold $M$, this action can be made isometric by constructing a metric on $M$ via a standard averaging process. In particular, if you have a (effective) circle action on $M$, you can find a metric $g$ on $M$ such that this action is isometric. In other words, the isometry group of $(M,g)$ contains a circle. This should provide you with plenty of examples: intuitively, a manifold that carries a circle action is one that has (at least) a "circle" worth of symmetries.


For example, take $M$ to be any rotationally symmetric surface in $\mathbb R^3$, e.g., a round sphere $S^2$ or a torus $S^1\times S^1$. Then $M$, by definition, has an isometric (effective) circle action. Moreover, these are the only (orientable) manifolds of dimension $2$ with an effective circle action. This follows from the fact that if we have a torus action on $M$, then the Euler characteristic of $M$ is $\chi(M)=\chi(M^T)$, where $M^T$ is the fixed point set of the torus. Since $M^T$ is a totally geodesic submanifold of $M$, it can be a circle or a finite collection of points. Thus, $\chi(M)\geq 0$, which means $M$ is homeomorphic to $S^2$, $\mathbb RP^2$, $S^1\times S^1$ or the Klein bottle $K^2$, according to orientability and $\chi(M)=2,1,0$, respectively. This is the classification you ask for in dimension equal to $2$.


Classifying compact connected manifolds of any dimension with a circle action and nothing else seems rather too general. More interesting problems are, for example, classifying manifolds with an isometric circle action that also have, e.g., some curvature condition. For example, if $\dim M=4$ and $M$ has a metric of positive curvature that has an isometric circle action, then [Hsiang-Kleiner](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214443064) proved that $M$ must be homeomorphic to $S^4$ or $\mathbb C P^2$.




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It might be worth pointing out a few **non-examples** as well; that is, manifolds that do not admit any effective $S^1$-actions (besides surfaces of genus $\geq 2$). It was proved by Borel that a closed aspherical manifold with centerless fundamental group does not admit nontrivial actions by a circle (hence any compact connected Lie group). Moreover, Atiyah and Hirzebruch showed that a closed spin manifold with non-vanishing $\hat A$-genus does not admit non-trivial circle actions.


