Editing Riemannian manifold (section)

== Every smooth manifold admits a Riemannian metric ==
'''Theorem:''' Every smooth manifold admits a (non-canonical) Riemannian metric.{{sfn|Lee|2018|p=11}}

This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed using only that a smooth manifold is a locally Euclidean topological space, for this result it is necessary to use that smooth manifolds are [[Hausdorff space|Hausdorff]] and [[paracompact]]. The reason is that the proof makes use of a [[partition of unity]]. 

{{Collapse top|title=Proof that every smooth manifold admits a Riemannian metric}}
Let <math> M</math> be a smooth manifold and <math>\{(U_\alpha,\varphi_\alpha)\}_{\alpha \in A}</math> a [[locally finite collection|locally finite]] [[atlas (topology)|atlas]] so that <math>U_\alpha \subseteq M</math> are open subsets and <math>\varphi_\alpha \colon U_\alpha\to \varphi_\alpha(U_\alpha)\subseteq\mathbf{R}^n</math> are diffeomorphisms. Such an atlas exists because the manifold is paracompact.

Let <math>\{\tau_\alpha\}_{\alpha \in A}</math> be a differentiable [[partition of unity]] subordinate to the given atlas, i.e. such that <math>\operatorname{supp}( \tau_\alpha) \subseteq U_\alpha</math> for all <math> \alpha \in A</math>.

Define a Riemannian metric <math>g</math> on <math>M</math> by
: <math>g = \sum_{\alpha \in A} \tau_\alpha \cdot \tilde{g}_\alpha</math>
where
:<math>\tilde{g}_\alpha = \varphi_\alpha^* g^{\text{can}}.</math>
Here <math>g^\text{can}</math> is the Euclidean metric on <math>\mathbb R^n</math> and <math>\varphi_\alpha^*g^{\mathrm{can}}</math> is its [[pullback (differential geometry)|pullback]] along <math>\varphi_\alpha</math>. While <math>\tilde{g}_\alpha</math> is only defined on <math>U_\alpha</math>, the product <math>\tau_\alpha \cdot \tilde{g}_\alpha</math> is defined and smooth on <math>M</math> since <math>\operatorname{supp}( \tau_\alpha) \subseteq U_\alpha</math>. It takes the value 0 outside of <math>U_\alpha</math>. Because the atlas is locally finite, at every point the sum contains only finitely many nonzero terms, so the sum converges. It is straightforward to check that <math>g</math> is a Riemannian metric.
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An alternative proof uses the [[Whitney embedding theorem]] to embed <math>M</math> into Euclidean space and then pulls back the metric from Euclidean space to <math>M</math>. On the other hand, the [[Nash embedding theorems#Ck embedding theorem|Nash embedding theorem]] states that, given any smooth Riemannian manifold <math>(M,g),</math> there is an embedding <math>F:M\to\mathbb{R}^N</math> for some <math>N</math> such that the [[pullback (differential geometry)|pullback]] by <math>F</math> of the standard Riemannian metric on <math>\mathbb{R}^N</math> is <math>g.</math> That is, the entire structure of a smooth Riemannian manifold can be encoded by a diffeomorphism to a certain embedded submanifold of some Euclidean space. Therefore, one could argue that nothing can be gained from the consideration of abstract smooth manifolds and their Riemannian metrics. However, there are many natural smooth Riemannian manifolds, such as the [[3D rotation group|set of rotations of three-dimensional space]] and hyperbolic space, of which any representation as a submanifold of Euclidean space will fail to represent their remarkable symmetries and properties as clearly as their abstract presentations do.