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Let $X,Y$ be length spaces, and suppose $f:X \to Y$ satisfies $$ (*) \, \, d^Y(f(x),f(y))<d^X(x,y)$$ for all $x,y \in X, x \neq y$.

Question: Is it possible that $f$ preserves lengths of paths?

Remark: Since length preserving is a local property, we can also localize the condition $(*)$, requiring it to only hold locally around every point of $X$.

(I am mainly interested in the smooth Riemannian case- when $X,Y$ are Riemannian manifolds, but then I do not assume $f$ is smooth or even $C^1$).

If there is such an $f$ (which preserves lengths), then it must not be a local homeomorphism. (Since an arcwise isometry that is a local homeomorphism is a local isometry).

Of course, if $f$ was $k$-Lipschitz for $k<1$ the answer would be negative.

Chill2Macht
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Asaf Shachar
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1 Answers1

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Example:

  • $X = (0,2\pi)$
  • $Y = \mathbb{R}^2$
  • $f(t)=(\cos(t),\sin(t))$
Lee Mosher
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