Why we have to definte covariant derivative?
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!
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 derivitive won't work.
Because the simple partial derivative of a tensor is not, in general, a tensor.
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 derivitive. See
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