- #1
noospace
- 75
- 0
Hi,
Let [itex]\mathbf{x}(u,v)[/itex] be a local parametrization of a regular surface. Then the coefficients of [itex]\mathbf{x}_{uu},\mathbf{x}_{uv}[/itex] etc. in the basis of the tangent space are defined as the Christoffel symbols.
On the other hand, if we write the first fundamental form [itex]\langle,\rangle[/itex] in differential form we have an extremization problem of the arc-length
[itex]ds^2 = E du^2 + 2F du dv + G dv^2[/itex].
Then the coefficients of of the corresponding Euler-Lagrange equations are essentially the Christoffel symbols.
Are there any interesting examples where the Lagrangian method of computing Christoffel symbols breaks down?
Let [itex]\mathbf{x}(u,v)[/itex] be a local parametrization of a regular surface. Then the coefficients of [itex]\mathbf{x}_{uu},\mathbf{x}_{uv}[/itex] etc. in the basis of the tangent space are defined as the Christoffel symbols.
On the other hand, if we write the first fundamental form [itex]\langle,\rangle[/itex] in differential form we have an extremization problem of the arc-length
[itex]ds^2 = E du^2 + 2F du dv + G dv^2[/itex].
Then the coefficients of of the corresponding Euler-Lagrange equations are essentially the Christoffel symbols.
Are there any interesting examples where the Lagrangian method of computing Christoffel symbols breaks down?