Semigeodesic parameterization I am a little confused

  • #1
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Today I was working through a handout my professor gave me on geodesics and stumbled upon the section on semigeodesic parameterization...and then I got lost.

I was able to follow the material through the point where the text points out that the first fundemental form will consist of:

I= E(du)^2 + G*(dv)^2, since the parametrization is orthogonal. The text then moves into an example of how if v were constant the differential equation for the geodesic

u''v'-u'v''+Av'-Bu' = 0 (where A and B's values can be derived from here: http://mathworld.wolfram.com/GeodesicCurvature.html )

becomes

-1/2*[tex]E_v[/tex]/G = 0.

Which makes sense, but then it gives an alternative statement later on that the above equation of the geodesic could also be expressed as

da/dv = - partial((G)^(1/2))/partial(u). Where a is defined as the angle the geodesics intersect the curves v=constant.

Whats worse is this alternative to the above was offered in the "theorems" section of the chapter and doesn't have a proof with it.

So two questions:

1) Wouldn't the angle be pi/2, since in a semigeodesic parameterization any geodesic would be orthogonal to the to the coordinate curves (in this case v=constant)?

2) Where are they getting the alternative expression...is is from Gauss-Bonnet? Or am I just reading the theorem incorrectly. In any case, where the heck is the text coming up with said "theorem." I don't want a proof, I just want a reasonable feel for where the heck they are getting it from. Because from their definations I just don't see the correlation.
 

Answers and Replies

  • #2
Reb
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Today I was working through a handout my professor gave me on geodesics and stumbled upon the section on semigeodesic parameterization...and then I got lost.

I was able to follow the material through the point where the text points out that the first fundemental form will consist of:

I= E(du)^2 + G*(dv)^2, since the parametrization is orthogonal. The text then moves into an example of how if v were constant the differential equation for the geodesic

u''v'-u'v''+Av'-Bu' = 0 (where A and B's values can be derived from here: http://mathworld.wolfram.com/GeodesicCurvature.html )

becomes

-1/2*[tex]E_v[/tex]/G = 0.

Which makes sense, but then it gives an alternative statement later on that the above equation of the geodesic could also be expressed as

da/dv = - partial((G)^(1/2))/partial(u). Where a is defined as the angle the geodesics intersect the curves v=constant.

Whats worse is this alternative to the above was offered in the "theorems" section of the chapter and doesn't have a proof with it.

So two questions:

1) Wouldn't the angle be pi/2, since in a semigeodesic parameterization any geodesic would be orthogonal to the to the coordinate curves (in this case v=constant)?

2) Where are they getting the alternative expression...is is from Gauss-Bonnet? Or am I just reading the theorem incorrectly. In any case, where the heck is the text coming up with said "theorem." I don't want a proof, I just want a reasonable feel for where the heck they are getting it from. Because from their definations I just don't see the correlation.


1) Not in the case of a random manifold.

2) Not really. It is derived by a change of variables in the pde [tex]E_v/2G=0[/tex].
 

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