Why we have to definte covariant derivative?

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Discussion Overview

The discussion revolves around the necessity of defining the covariant derivative in the context of differential equations on manifolds. Participants explore the limitations of using partial derivatives and other derivatives like the Lie derivative when dealing with tensors and vector fields on curved surfaces.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions the need for a covariant derivative, suggesting it relates to understanding differential equations on manifolds.
  • Another participant notes that simple partial derivatives of tensors do not generally yield tensors, implying a limitation in their use.
  • A further explanation is provided regarding the intrinsic nature of derivatives along curves on surfaces, emphasizing that the covariant derivative captures the projection onto the manifold.
  • There is a clarification about a possible misunderstanding of the term "definte," which may indicate a focus on the importance of definitions in mathematics.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding and reasoning regarding the covariant derivative, but there is no consensus on a singular explanation or necessity. Multiple viewpoints on the topic remain present.

Contextual Notes

The discussion does not resolve the underlying assumptions about the nature of tensors or the specific conditions under which different derivatives are applicable.

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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 derivative 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 derivative. See

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