Why we have to definte covariant derivative?

[HeilPhysicsPhysics]

Why we have to definte covariant derivative?

 

[Data]

I don't know what "definte" means, but if you mean "define," then, well, you have to define things in order to know what they are!

 

[Daverz]

I think the question is probably why, when writing down differential equations on fields on a manifold we can't we just use partial derivitives. And now I'm trying to think why the Lie derivative won't work.

 

[HallsofIvy]

Because the simple partial derivative of a tensor is not, in general, a tensor.

 

[Daverz]

A helpful way to think about it is that if you had a curve with parameter t on a surface and vector field V along along the curve and tangent to the surface, dV/dt would not be intrinsic to the surface, but it's projection onto the manifold would be, and this is precisely the covariant derivative. See

http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/Sec8.html