drgigi
Sep15-10, 07:09 PM
Hi!
I've finally decided to tackle a diff geom book, but I'm having trouble with this Problem 4/Chapter 2 from Do Carmo's Riemannian Geometry:
Let M^2\subset R^3 be a surface in R^3 with induced Riemannian metric. Let c:I\rightarrow M be a differentiable curve on M and let V be a vector field tangent to M along c; V can be thought of as a smooth function V:I\rightarrow R^3, with V(t)\in T_{c(t)}M.
a)show that V is parallel if and only if dV/dt is perpendicular to T_{c(t)}\subset R^3 where dV/dt is the usual derivative of V:I\rightarrow R^3
b) hopefully I can handle myself. will come back if not! :)
So I guess the plan is to use
DV/dt=(dv^k/dt + \Gamma^k_{ij} v^j dx^i/dt) X_k=0
and dot it with some vector u^iX_i. If I can show that the second term in Dv/dt dotted with this u is zero the problem is done, but I don't see why that should be true..
if I dot X_i with X_j i get \delta_{i,j}, right? what then?
any hints would be great! Thanks!
I've finally decided to tackle a diff geom book, but I'm having trouble with this Problem 4/Chapter 2 from Do Carmo's Riemannian Geometry:
Let M^2\subset R^3 be a surface in R^3 with induced Riemannian metric. Let c:I\rightarrow M be a differentiable curve on M and let V be a vector field tangent to M along c; V can be thought of as a smooth function V:I\rightarrow R^3, with V(t)\in T_{c(t)}M.
a)show that V is parallel if and only if dV/dt is perpendicular to T_{c(t)}\subset R^3 where dV/dt is the usual derivative of V:I\rightarrow R^3
b) hopefully I can handle myself. will come back if not! :)
So I guess the plan is to use
DV/dt=(dv^k/dt + \Gamma^k_{ij} v^j dx^i/dt) X_k=0
and dot it with some vector u^iX_i. If I can show that the second term in Dv/dt dotted with this u is zero the problem is done, but I don't see why that should be true..
if I dot X_i with X_j i get \delta_{i,j}, right? what then?
any hints would be great! Thanks!