Intuition behind riemannian metric

Question

I formally understand what a riemannian metric $g$ on a manifold $M$ is. It's basically a section of the vectorbundle $T^*M\otimes T^*M \to M$ (which in the end corresponds to mapping 2 vectors from a tangent space at a base point to a real number after writing it out), that's symmetric, and positive definite. Bilinearity follows from this, therefor the riemann metric $g$ is basically assigning an inner product to each tangent space.

However, I do not have an intuition of the total. Simply having an inner product on each tangent space is understandable, but not the total structure has to be smooth, aka we know that sections of vectorbundles have to be smooth. I don't have a geometric intuition for this. For example for a vector field we have that at each point you have a different tangent vector, but the smoothness in this case basically says that you don't have "strange jumps or whatever" for the vectorfield (basically the flow is smooth). How about the riemannian metric, is there a similar analogy, and could it be explained?

Answer 1

Well locally in coordinates its a function from the manifold to a space of matrices. Then its obvious what it means to vary smoothly.

Answer 2

To say the metric $g$ is smooth means that if you feed it with any two smooth vector fields $X,Y$, the map $g(X,Y):M\to\mathbb{R}$ is again smooth.

Indeed, that's how the notion of smoothness is inherited to all the tensor powers of the tangent (and cotangent) bundle:

Take a smooth $n$-dimensional manifold M. The smooth structure is given by an atlas $\mathscr{A}_M$ consisting of $C^{\infty}$-compatible charts.

If a map $f:M\to\mathbb{R}$ is such that for every chart $(U,u)\in\mathscr{A}_M$ the composition $f\circ u^{-1}:\mathbb{R}^{n}\to\mathbb{R}$ is smooth, then we call $f$ smooth, and we call $C^{\infty}$ the space of all such maps.

Then (after defining what tangent spaces are, what the tangent bundle $TM$ is, and what a section is) we say a section $X:M\to TM$ is smooth if for every $f\in C^{\infty}(M)$ the map $Xf:M\to\mathbb{R}$ defined by $(Xf)_p:=X_pf$ is again smooth (an element of $C^{\infty}(M)$). We call this space $\Gamma(TM)$.

We define smooth covector fields to be the sections of $T^{*}M\to M$ that take any smooth vector field into a smooth map $M\to\mathbb{R}$. We call this space $\Gamma(T^{*}M)$.

And so on.