We learned that two paths are homotopic if they can be continuosly transformed into each other by keeping their start and endpoints fixed. Does that always mean that two homotopic paths have the same length?
2026-05-14 12:35:41.1778762141
Do 2 homotopic paths always have the same lenght?
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First of all, in general topological spaces you have no notion of length so you can not speak about it. If you assume that you are working on $\mathbb{R}^n$, then it makes sense to ask that question and the response is negative. See for example the image taken from Google (we imagine the ambient space is $\mathbb{R}^2$ :
All the paths are homotopic despite of their lengths are distinct.
If you want to convince yourself computing the lengths I recommend you to have a look to Martin's Crossley's book: Essential Topology. In the introduction to homotopy he presented two formulas for a family of "simple"(easy to manipulate and understant) functions (polynomials) which are homotopic (in $\mathbb{R}$) but as you can compute with a few integrals the lengths are different.
Hope that helps.