Length shortening Riemannian metrics

Question

I am looking for examples of Riemannian metrics such that the curve length under these metrics are always smaller than the length as measured in Euclidean space. It is just a question that popped into my head today. However, browsing the internet does not suggest anything. Thanks in advance

Answer 1

Well, let $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ be a shrinking map, for example $\varphi(x)=x/2$. For every embedded manifold $M\subset\mathbb{R}^n$, the restricted map $\varphi|_M$ is another embedding of $M$ in the Euclidean space. Now just take the pullback Riemannian metric $\varphi^*g$ on $M$, where $g$ is the standard metric on $\mathbb{R}^n$.

Another way to express the same idea is the following. Given a Riemannian manifold $(M,g)$, one can define a new Riemannian metric by $g'=a\cdot g$, for any $a>0$. In fact, one can even set $g'=f\cdot g$, where $f$ is a positive smooth function on $M$. If $f$ is everywhere $<1$, you'll get what you want.