0

This might be a basic question, but it's been irking me for the past few days.

The common definition of a manifold is as a second-countable, Haussdorff topoplogical space which is locally homeomorphic to a Euclidean space. One can then impose a smooth structure and turn it into a smooth manifold. Furhter, one may equip a smooth manifold with an inner product on each tangent space (smoothly varying of course) and obtain a Riemannian manifold.

If my understanding is correct, a Euclidean space comes with an associated inner product on its corresponding vector space. Do we really need this inner product? In my mind, a smooth manifold should be to a Riemannian manifold as an affine space is to a Euclidean space, since we add an inner product in both cases. Do we really need the inner product in the definition of a manifold? Can't we just say it's locally homeomorpic to an affine space?

markusas
  • 358
  • 1
    I don't follow. There is no inner product in the definition of manifold. Of course you can replace the Euclidean space in the definition of manifold with the affine space of the same dimension, or with any other space homeomorphic to it. Homeomorphisms don't see inner products, or metrics, or other such features. To come at it from a different angle, when $\mathbb{R}^n$ comes up in topology it usually is not as an inner product space, but simply as a nice topological space or perhaps topological vector space. – Ben Steffan Jun 05 '24 at 12:14
  • @BenSteffan I see. I don't mean to say that an inner product is in the definition of a manifold. Euclidean space is in the definition though, whose associated vector space has an inner product. But your claim that Euclidean space in the definition of a manifold can be replaced with affine space answers my question. Is there a specific reason we don't usually define it this way? Just by convention/convenience? – markusas Jun 05 '24 at 13:31
  • 1
    See my answer here or Lang’s book. https://math.stackexchange.com/a/4927033/10584 – Deane Jun 05 '24 at 13:32
  • 1
    @markusas At the danger of repeating myself (somewhat), in topology "Euclidean space" first and foremost means "the topological space $\mathbb{R}^n$." Not a vector space, not an inner product space, just a topological space. We use all these additional structures sometimes, but they are understood as properties of the topological space and are secondary to it. As to why we use $\mathbb{R}^n$ over $\mathbb{A}^n$: this convention is inherited from analysis, and you don't see $\mathbb{A}^n$ around in analysis all that much. – Ben Steffan Jun 05 '24 at 14:59
  • 1
    Note that you can say "locally homeomorpic to an affine space" only if affine space has a topology. The easiest way to define the standard topology of affine space is to put a Euclidean structure on the affine space and use the inner product. You can then show that the topology is independent of the inner product used. – Deane Jun 05 '24 at 20:48

0 Answers0