Transformation on surface patches that preserves length of curves.

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suppose $mathcalS subset mathbbR^3$ is a surface patch, and let $mathcalC$ a simple curve on $mathcalS$, suppose $f$ is a transformation/function/mapping such that $f(mathcalS) = mathcalS'$ with the property that for any $mathcalC$ we have $length(f(mathcalC)) = length(C)$.



Folding a piece of paper for example I guess would have this property.



But anyway, I'm not an expert in differential geometry but do these transformations here have a name, is there any study about transformations that have such property?



I need some name for references so I can look it up.



Thank you



(Again, I'm not an expert in differential geometry, but I do know "elementary differential geometry").







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  • Such a map is called a (local) isometry. Actually the definition of an isometry is different, but your assumption implies the the map is an isometry (locally).
    – Thomas
    Aug 8 at 18:09















up vote
0
down vote

favorite












suppose $mathcalS subset mathbbR^3$ is a surface patch, and let $mathcalC$ a simple curve on $mathcalS$, suppose $f$ is a transformation/function/mapping such that $f(mathcalS) = mathcalS'$ with the property that for any $mathcalC$ we have $length(f(mathcalC)) = length(C)$.



Folding a piece of paper for example I guess would have this property.



But anyway, I'm not an expert in differential geometry but do these transformations here have a name, is there any study about transformations that have such property?



I need some name for references so I can look it up.



Thank you



(Again, I'm not an expert in differential geometry, but I do know "elementary differential geometry").







share|cite|improve this question




















  • Such a map is called a (local) isometry. Actually the definition of an isometry is different, but your assumption implies the the map is an isometry (locally).
    – Thomas
    Aug 8 at 18:09













up vote
0
down vote

favorite









up vote
0
down vote

favorite











suppose $mathcalS subset mathbbR^3$ is a surface patch, and let $mathcalC$ a simple curve on $mathcalS$, suppose $f$ is a transformation/function/mapping such that $f(mathcalS) = mathcalS'$ with the property that for any $mathcalC$ we have $length(f(mathcalC)) = length(C)$.



Folding a piece of paper for example I guess would have this property.



But anyway, I'm not an expert in differential geometry but do these transformations here have a name, is there any study about transformations that have such property?



I need some name for references so I can look it up.



Thank you



(Again, I'm not an expert in differential geometry, but I do know "elementary differential geometry").







share|cite|improve this question












suppose $mathcalS subset mathbbR^3$ is a surface patch, and let $mathcalC$ a simple curve on $mathcalS$, suppose $f$ is a transformation/function/mapping such that $f(mathcalS) = mathcalS'$ with the property that for any $mathcalC$ we have $length(f(mathcalC)) = length(C)$.



Folding a piece of paper for example I guess would have this property.



But anyway, I'm not an expert in differential geometry but do these transformations here have a name, is there any study about transformations that have such property?



I need some name for references so I can look it up.



Thank you



(Again, I'm not an expert in differential geometry, but I do know "elementary differential geometry").









share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 8 at 17:01









user8469759

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  • Such a map is called a (local) isometry. Actually the definition of an isometry is different, but your assumption implies the the map is an isometry (locally).
    – Thomas
    Aug 8 at 18:09

















  • Such a map is called a (local) isometry. Actually the definition of an isometry is different, but your assumption implies the the map is an isometry (locally).
    – Thomas
    Aug 8 at 18:09
















Such a map is called a (local) isometry. Actually the definition of an isometry is different, but your assumption implies the the map is an isometry (locally).
– Thomas
Aug 8 at 18:09





Such a map is called a (local) isometry. Actually the definition of an isometry is different, but your assumption implies the the map is an isometry (locally).
– Thomas
Aug 8 at 18:09
















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