- #1
MathematicalPhysicist
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I might have forgotten about it cause I took a similar course two years ago.
So I have this assertion:
Let F be a [tex]C^1[/tex] function from R^n to R^n, show that if F is injective, and for each curve [tex]\gamma : I\rightarrow R^n[/tex] [tex]Length(\gamma)=Length(F o \gamma)[/tex] then F is an isometry.
So I thought basically if I pick [tex]\gamma[/tex] to be a straight line, and if the metric is the euclidean metric then basically I have [tex]Length(\gamma)=d(\gamma(a),\gamma(b))[/tex]
and from what is given I get that the metric is conserved as well under F, so it's an isometry.
But I am not sure I should use here a specific metric.
Any other thought of this question?
Thanks.
So I have this assertion:
Let F be a [tex]C^1[/tex] function from R^n to R^n, show that if F is injective, and for each curve [tex]\gamma : I\rightarrow R^n[/tex] [tex]Length(\gamma)=Length(F o \gamma)[/tex] then F is an isometry.
So I thought basically if I pick [tex]\gamma[/tex] to be a straight line, and if the metric is the euclidean metric then basically I have [tex]Length(\gamma)=d(\gamma(a),\gamma(b))[/tex]
and from what is given I get that the metric is conserved as well under F, so it's an isometry.
But I am not sure I should use here a specific metric.
Any other thought of this question?
Thanks.