Geodesic curvature, normal curvature, and geodesic torsion

[Demon117]

I am struggling to make sense out some things. Hopefully someone can help or at least offer some different point of view. Let's examine a differential curve parameterized by arc length that maps some interval into an oriented surface (lets call it N(s)). The surface has a unit normal field restricted to the curve $\alpha$. Also, let $n(s):=N(s)\wedge T(s)$ where $T(s)=\alpha '(s)$.

If we define the derivatives of T, N, and n as the following

$T'=-k_{g}n+k_{n}N$
$n'=k_{g}T+\tau_{g}N$
$N'=-k_{n}T-\tau_{g}n$

then we should have $N'\cdot T = -k_{n}$ and the second fundamental form is given by $II(T,T) = k_{n}$ while $N'\cdot n=-\tau_{g}$ so that the second fundamental form is given by $II(T,n)=\tau_{g}$.

This seems pretty clear to me, unless I have my definitions mixed up some how. Does this seem correct?

 

[micromass]

I am struggling to make sense out some things. Hopefully someone can help or at least offer some different point of view. Let's examine a differential curve parameterized by arc length that maps some interval into an oriented surface (lets call it N(s)). The surface has a unit normal field restricted to the curve $\alpha$. Also, let $n(s):=N(s)\wedge T(s)$ where $T(s)=\alpha '(s)$.

Hmm. I'm not sure what your $N(s)$ is supposed to be. Is $N(s)$ the normal of the surface at point $\alpha(s)$?

If we define the derivatives of T, N, and n as the following

$T'=-k_{g}n+k_{n}N$
$n'=k_{g}T+\tau_{g}N$
$N'=-k_{n}T-\tau_{g}n$

I get some sign differences. I think you should see http://en.wikipedia.org/wiki/Darboux_frame for the correct definitions.

the second fundamental form is given by $II(T,T) = k_{n}$ while $N'\cdot n=-\tau_{g}$ so that the second fundamental form is given by $II(T,n)=\tau_{g}$.

Not sure where this comes from. Could you clarify?

 

[Demon117]

Hmm. I'm not sure what your $N(s)$ is supposed to be. Is $N(s)$ the normal of the surface at point $\alpha(s)$?

Yes that is correct. I made a mistake in my initial statement. N(s) is the unit normal field to the surface at $\alpha(s)$. Thanks for allowing me to clarify this. That means n(s) is the normal to the curve $\alpha$.

I get some sign differences. I think you should see http://en.wikipedia.org/wiki/Darboux_frame for the correct definitions.

I will look into this.

Not sure where this comes from. Could you clarify?

The second fundamental form defined by $II(v,v)=-\left\langle dN_{p}v,v\right\rangle$ for any v in the tangent plane at a point p of S.

 

[micromass]

The second fundamental form defined by $II(v,v)=-\left\langle dN_{p}v,v\right\rangle$ for any v in the tangent plane at a point p of S.

I agree with that then, up to signs. (Not that signs are all that important)

 

[Demon117]

I agree with that then, up to signs. (Not that signs are all that important)

That is interesting, because I spoke with a professor recently about this and his claim was that the geodesic torsion was defined by $II(T,N)=\tau_{g}$. But by inspection this wouldn't make sense. I'm just confused a little I guess.

 

[micromass]

That is interesting, because I spoke with a professor recently about this and his claim was that the geodesic torsion was defined by $II(T,N)=\tau_{g}$. But by inspection this wouldn't make sense. I'm just confused a little I guess.

Signs are not so important anyway. If he defines his geodesic torsion like that, then I don't think it'll make a lot of difference. It'll yield the same theory up to sign.

 

[Demon117]

Signs are not so important anyway. If he defines his geodesic torsion like that, then I don't think it'll make a lot of difference. It'll yield the same theory up to sign.

I agree with the sign issue but I think it does make a huge difference because $N'\cdot n \ne N'\cdot N$ by definition of the Darboux frame, and by that same definition $N'\cdot n=\tau_{g}$. So $II(T,N)\ne \tau_{g}$.