Why we have to definte covariant derivative?

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Defining the covariant derivative is essential for understanding how to differentiate tensors on manifolds, as simple partial derivatives do not yield tensors in general. The covariant derivative accounts for the curvature of the manifold, allowing for intrinsic differentiation along curves on the surface. Unlike the Lie derivative, which does not maintain tensorial properties, the covariant derivative ensures that the results remain consistent with the manifold's geometry. This is crucial for formulating differential equations in the context of fields on manifolds. Understanding these concepts is vital for advanced studies in differential geometry and theoretical physics.
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Why we have to definte covariant derivative?
 
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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

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