Turning tangent theorem

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Someone knows where can I find the proof of the following theorem?




Let be $S$ a surface in $mathbbR^3$ and $R subset S$ a simply connected region. Let be $(e_1, e_2, e_3)$ an adapted frame defined on R (which changes as $P in R$ changes). Let be $phi$ the 1-form which measures the "infinitesimal angle" (or the change of angle) between $e_1$ and the unit tangent vector of $partial ^+R$. Then



$int _partial ^+Rphi = 2pi$




I know that it is similar to the theorem for plane curves, but there you have a fixed frame of $mathbbR^2$, here you have a moving frame on the surface.



Thanks in advance




differential-geometry differential-topology differential-forms




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  • Does "adapted frame" mean $TR = mathrmspan(e_1,e_2)?$
    – Anthony Carapetis
    Jul 16 at 11:18










  • Yes! And $e_3$ normal to surface. And ortonormal
    – Marco All-in Nervo
    Jul 16 at 12:38














up vote
0
down vote

favorite












Someone knows where can I find the proof of the following theorem?




Let be $S$ a surface in $mathbbR^3$ and $R subset S$ a simply connected region. Let be $(e_1, e_2, e_3)$ an adapted frame defined on R (which changes as $P in R$ changes). Let be $phi$ the 1-form which measures the "infinitesimal angle" (or the change of angle) between $e_1$ and the unit tangent vector of $partial ^+R$. Then



$int _partial ^+Rphi = 2pi$




I know that it is similar to the theorem for plane curves, but there you have a fixed frame of $mathbbR^2$, here you have a moving frame on the surface.



Thanks in advance




differential-geometry differential-topology differential-forms




share|cite|improve this question



















  • Does "adapted frame" mean $TR = mathrmspan(e_1,e_2)?$
    – Anthony Carapetis
    Jul 16 at 11:18










  • Yes! And $e_3$ normal to surface. And ortonormal
    – Marco All-in Nervo
    Jul 16 at 12:38












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Someone knows where can I find the proof of the following theorem?




Let be $S$ a surface in $mathbbR^3$ and $R subset S$ a simply connected region. Let be $(e_1, e_2, e_3)$ an adapted frame defined on R (which changes as $P in R$ changes). Let be $phi$ the 1-form which measures the "infinitesimal angle" (or the change of angle) between $e_1$ and the unit tangent vector of $partial ^+R$. Then



$int _partial ^+Rphi = 2pi$




I know that it is similar to the theorem for plane curves, but there you have a fixed frame of $mathbbR^2$, here you have a moving frame on the surface.



Thanks in advance




differential-geometry differential-topology differential-forms




share|cite|improve this question











Someone knows where can I find the proof of the following theorem?




Let be $S$ a surface in $mathbbR^3$ and $R subset S$ a simply connected region. Let be $(e_1, e_2, e_3)$ an adapted frame defined on R (which changes as $P in R$ changes). Let be $phi$ the 1-form which measures the "infinitesimal angle" (or the change of angle) between $e_1$ and the unit tangent vector of $partial ^+R$. Then



$int _partial ^+Rphi = 2pi$




I know that it is similar to the theorem for plane curves, but there you have a fixed frame of $mathbbR^2$, here you have a moving frame on the surface.



Thanks in advance





differential-geometry differential-topology differential-forms





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 15 at 22:55









Marco All-in Nervo

1128




1128











  • Does "adapted frame" mean $TR = mathrmspan(e_1,e_2)?$
    – Anthony Carapetis
    Jul 16 at 11:18










  • Yes! And $e_3$ normal to surface. And ortonormal
    – Marco All-in Nervo
    Jul 16 at 12:38
















  • Does "adapted frame" mean $TR = mathrmspan(e_1,e_2)?$
    – Anthony Carapetis
    Jul 16 at 11:18










  • Yes! And $e_3$ normal to surface. And ortonormal
    – Marco All-in Nervo
    Jul 16 at 12:38















Does "adapted frame" mean $TR = mathrmspan(e_1,e_2)?$
– Anthony Carapetis
Jul 16 at 11:18




Does "adapted frame" mean $TR = mathrmspan(e_1,e_2)?$
– Anthony Carapetis
Jul 16 at 11:18












Yes! And $e_3$ normal to surface. And ortonormal
– Marco All-in Nervo
Jul 16 at 12:38




Yes! And $e_3$ normal to surface. And ortonormal
– Marco All-in Nervo
Jul 16 at 12:38















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