The term "metric" is familiar, but not the idea of a pullback on it. I have tried to find intuitive, beginner-friendly explanations of this concept without success. Your attempts would be appreciated. Pictures and concrete examples would be wonderful, if possible.
I have not studied much topology or differential geometry before, but know some really early engineering/physics math (linear algebra, multivariate and vector calculus etc.) Analogies to these areas would be great.
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$\begingroup$Suppose that you have two spaces $X$ and $Y,$ a metric $d$ on $Y$, and a function $f : X \to Y.$ The pullback metric is the following metric on $X$: $$(f^*d)(x^{(1)}, x^{(2)}) = d(f(x^{(1)}), f(x^{(2)})); \quad x^{(1)}, x^{(2)} \in X$$
Thus, we define a metric on $X$ by mapping points over to $Y$ and taking the distance there.
One example is given by considering different coordinate systems. Let $E = \mathbb R^2$ be the plane with ordinary Euclidean distance, $$d_E((x_1, y_1), (x_2, y_2)) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}.$$
Let $P = [0, \infty) \times [0, 2\pi),$ the domain of polar coordinates. Define $f : P \to E$ by $f(r, \theta) = (r \cos\theta, r \sin\theta).$ You probably recognize this as the mapping from polar coordinates to Cartesian coordinates. The pullback metric $d_P := f^*d_E$ is then $$d_P((r_1, \theta_1), (r_2, \theta_2)) = \sqrt{(r_1 \cos\theta_1 - r_2 \cos\theta_2)^2 + (r_1 \sin\theta_1 - r_2 \sin\theta_2)^2.}$$
This is the distance between two points in a plane, given in polar coordinates.
I believe that it is called pullback since we pull the metric from the codomain of $f$ back to the domain of $f$.
$\endgroup$ 1 $\begingroup$If you have a metric $d_Y$ on $Y$ you can pull it back to $X$ via $f:X\to Y$ by setting $d_X(a,b)$ to be $d_Y(f(a),f(b))$.
If $f$ is injective you get an honest metric, but otherwise the “metric” fails the non degeneracy requirement
This is completely analogous to the case of inner products in linear algebra.
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