Why Riemannian metrics have to be smooth?

Question

Why do Riemannian metrics have to be smooth? Can you give an example of a smooth curve with a none smooth metric and show me what possibly will go wrong if our metric is not smooth?

Answer 1

To begin with, nothing has to be smooth. There are deep theories concerning manifolds which are only $C^k$ for some $k$, or topological manifolds, which don't even have to be differentiable.

But when considering the category of smooth manifolds and smooth maps, certain constructions have to be smooth in order for the resulting objects to fit into this category. For example, the inverse image of a regular value under a smooth map is again a smooth manifold. If we omit the requirement that the initial map is smooth, the resulting manifold will not be smooth either (actually, sometimes it will not be a manifold at all). Alternatively, consider smooth vector fields. We know that such a vector field has a flow function which induces diffeomorphisms from the given manifold to itself. If the vector field is not smooth, the resulting automorphisms will not be smooth either, thus will not preserve the basic structure of any object in this category.

It is the same with the metric tensor. This tensor actually gives an isomorphism between the tangent and cotangent spaces at every point on the manifold. When it is also smooth, it yields an isomorphism of vector bundles between the tangent and cotangent bundles, i.e. it transforms smooth $1$-forms into smooth vector fields, and vice versa. Just one example, out of many - Let $f:M\to\mathbb{R}$ be a smooth function, so the derivative $df$ is a smooth $1$-form, and the metric induces the corresponding vector field $\nabla f$, the gradient. If the metric is smooth, then so is $\nabla f$, thus it has a smooth flow function and so on. If the metric is not smooth, then neither is the obtained gradient, and it just means that this vector field does not fit into the smooth category.

There are many other examples for things that "go wrong" if the metric is not smooth. The "bad" outcome is always that some obtained map or manifold or any other object will not be smooth either. It is worth mentioning that this "bad" outcome is not always so bad. As written above, there is a lot to do with non-smooth constructions. However, there are some results which require smoothness or at least the existence of many derivatives, and these results may fail to hold if one uses a non-smooth metric.