Differential Topology – Why There is No Natural Metric on Manifolds

Question

One of the things that always bothered me after learning introductory differential geometry (as a physics student) and then delving deeper into this field on my own is that, the usual construction of differential manifolds are such that $M$ is an $n$-dimensional manifold if it is 1) a Hausdorff-space, 2) it is locally homeomorphic to $\mathbb{R}^{n}$, 3) the transistion functions between charts are invertable and continuous/differentiable/analytic etc. depending on manifold type.

Now as far as I understand, norm, metric, etc. are topological properties and homeomorphisms are maps that preserve topological structure. Yet, metric is an extra structure on a manifold, not part of a manifold's given properties. But if manifolds are locally homeomorphic to a set that has a very strong topological structure, including a metric ($\mathbb{R}^{n}$) then why is that structure not preserved locally?

Secondly, which is probably related to this, if the transition functions are differentiable, as it is demanded for differential manifolds, does that mean that the coordinate maps themselves are also differentiable?
Because then they are basically diffeomorphisms from $M$ to $\mathbb{R}^{n}$. And if it is so, then given the natural metric tensor $g$ on $\mathbb{R}^{n}$, why can't that metric be induced on the manifold via the pullback?

Answer 1

First, I think you have some misconceptions about what it means to be a "topological property." A topological property is a property that is preserved by homeomorphisms. Norms and metrics are definitely not topological properties. For example, the unit ball in $\mathbb R^n$ is homeomorphic (in fact, diffeomorphic) to $\mathbb R^n$ itself, but the two spaces have very different metric properties. (One is bounded and the other is not, for example.) You can certainly use a metric to induce a topology; but many different metrics will induce the same topology.

Second, if $M$ is a smooth manifold and $\phi$ is a coordinate map from an open set $U\subseteq M$ to an open set $\widehat U\subseteq\mathbb R^n$, then yes, $\phi$ is a diffeomorphism from $U$ to $\widehat U$.

Third, given a coordinate map $\phi$ as above, you can certainly use the pullback by $\phi$ to induce a metric on $U$. But note that this metric will only be defined on the subset $U$, typically not on all of $M$, and different coordinate charts will induce different metrics. That's why a metric is an extra piece of data that has to be chosen; it's not determined intrinsically by the smooth manifold structure.

EDIT: Your comment suggests that you're thinking of "Riemannian" as a property that a smooth manifold might or might not have. It's not; instead, it's an additional layer of structure that we can add to a smooth manifold if we wish.

In fact, there are four different layers of structure here. Let's take $\mathbb S^2 = \{(x,y,z)\in\mathbb R^3: x^2 + y^2 + z^2 = 1\}$ as an example. It has the following four structures:

1. It's a set: By definition, $\mathbb S^2$ is just a certain set of ordered triples of real numbers.
2. We can make it into a topological space: By specifying a topology on $\mathbb S^2$ (which is just a certain collection of subsets satisfying certain conditions), we make it into a topological space. Usually, when talking about a subset of a Euclidean space, we give it the subspace topology: that is, we declare a subset of $\mathbb S^2$ to be open iff it is the intersection of $\mathbb S^2$ with an open subset of $\mathbb R^3$. Note that this topology was originally defined in terms of a metric (the Euclidean distance function), but we're not going to use that metric for anything other than deciding which sets are open. Once we've decided on a topology, we can note that this topology happens to have the properties that make it a topological manifold: it's Hausdorff, second-countable, and locally Euclidean.
3. Then we can make it into a smooth manifold: Note that "smoothness" is not a topological property. If it were, we'd certainly want to say that $\mathbb S^2$ is smooth; but $\mathbb S^2$ is homeomorphic to the surface of a cube, which we would probably not want to say is smooth. Instead, a "smooth structure" is an additional piece of structure that we have to put on a manifold: it's a choice of a particular collection of coordinate charts that overlap smoothly. Fortunately, because of the way $\mathbb S^2$ sits in $\mathbb R^3$, there's a natural choice of smooth structure on it, and that's generally the one we use (often without mentioning that there's a choice).
4. Then we can make it into a Riemannian manifold: "Riemannian" is not a property that a smooth manifold might or might not have. Instead, a Riemannian metric is a choice of inner product on each tangent space, in such a way that it varies smoothly from point to point. For $\mathbb S^2$, a very natural choice is to declare the inner product on each tangent space to be the restriction of the Euclidean dot product. But we don't have to choose this one. For example, let $M$ be a smooth hot-dog-shaped surface in $\mathbb R^3$, endowed with the Riemannian metric obtained by restricting the Euclidean dot product. There's a diffeomorphism $F\colon \mathbb S^2\to M$, and we could also choose to give $\mathbb S^2$ the metric obtained by pulling back $M$'s metric via $F$. How you define the metric depends on what your purposes are.

One reason it's easy to get confused about these layers of structure is that when there are obvious "natural" choices such as the ones I described above, we often don't even mention that a choice is being made. For example, if an author writes "Let $\mathbb S^2$ be the unit sphere in $\mathbb R^3$," you have to decide from the context whether she's thinking of it as a set, or a topological space with the subspace topology, or a smooth manifold with the induced smooth structure, or as a Riemannian manifold with the induced Riemannian structure.

You can find lots more detail about these things in Chapter 1 of my book Introduction to Smooth Manifolds.