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}.