Some mathematical structures are visualized very well. I imagine how a shapeless bunch of points (a set; the only property of which is quantity) is collected in one or another soft form (topological space), depending on what topology is set on it. I imagine how this soft (floating) shape becomes hard when I set the metric. The first examples of deformation retractions give a vivid geometric intuition of the homotopy type as a "(homotopy) framework". Thus, the concepts of homotopy type, topology, and metrics have a clear physical content. In contrast to these three examples, for the concept of a smooth structure on a manifold, everything is not so clear due to the fact that manifolds up to dimension 3 have a unique smooth structure

Do you have any intuition for a smooth structure on a manifold? Do you see any meaning/physical content in it? Do visually homeomorphic non-diffeomorphic spaces differ for you?

After a smooth manifold is provided with additional structures (Riemannian, symplictic, etc.), the meaning of these objects becomes clear. But a smooth manifold devoid of additional structures is still very mysterious for me.

P.S. I'm not sure if this question is for a forum (on the other hand, the system showed me many questions with similar titles). If so, then feel free to close it.

Some mathematical structures are visualized very well. I imagine how a shapeless bunch of points (a set; the only property of which is quantity) is collected in one or another soft form (topological space), depending on what topology is set on it. I imagine how this soft (floating) shape becomes hard when I set the metric. The first examples of deformation retractions give a vivid geometric intuition of the homotopy type as a "(homotopy) framework". Thus, the concepts of homotopy type, topology, and metrics have a clear physical content. In contrast to these three examples, for the concept of a smooth structure on a manifold, everything is not so clear due to the fact that manifolds up to dimension 3 have a unique smooth structure

Do you have any intuition for a smooth structure on a manifold? Do you see any meaning/physical content in it? Do visually homeomorphic non-diffeomorphic spaces differ for you?

After a smooth manifold is provided with additional structures (Riemannian, symplectic, etc.), the meaning of these objects becomes clear. But a smooth manifold devoid of additional structures is still very mysterious for me.

P.S. I'm not sure if this question is for a forum (on the other hand, the system showed me many questions with similar titles). If so, then feel free to close it.