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I am trying to better understand the concept of connection in diff. geometry by defining different vectors fields and working with them. For example, if I take the usual coordinates $x$ and $y$ coordinates on the plane, I have the vector fields $\partial_x$ and $\partial_y$.

Now if I replace the coordinate $y$ by $z=y+x^2$, then I should replace $\partial_y$ with $\partial_z=\partial_y+2x\partial_x$.

On the one hand, I think the pair $\{\partial_x,\partial_z\}$ can still be used as a basis for the tangent space at every point and this basis is holonomic, i.e. the elements are derivatives with respect to coordinates.

On the other hand, holonomic basis are supposed to be commutative, but I get $[\partial_x,\partial_z]\neq 0$. Can someone clarify this, please?

thedude
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1 Answers1

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This is easily fixed. We just need to be more careful when we apply the chain rule.

To avoid confusion, I'll introduce a new coordinate $w = x$, so the old coordinates are $(x,y)$ and the new coordinates are $$(w(x,y),z(x,y)) = (x, y + x^2).$$ Inverting this, we can write the old coordinates in terms of the new coordinates:$$(x(w,z),y(w,z)) = (w,z - w^2)$$

The chain rule says that $$ \partial_w = \frac{\partial x}{\partial w}\partial_x + \frac{\partial y}{\partial w}\partial_y = \partial_x-2w\partial_y=\partial_x-2x\partial_y, \\ \partial_z = \frac{\partial x}{\partial z}\partial_x + \frac{\partial y}{\partial z}\partial_y = \partial_y. $$

It should now be easy to see that $ [\partial_w, \partial_z] = 0$, which is what you want.

Kenny Wong
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