Definition of differentiability on a manifold

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quasar987
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My text defines differentiability of [itex]f:M\rightarrow \mathbb{R}[/itex] at a point p on a manifold M as the differentiability of [itex]f\circ \phi^{-1}:\phi(V) \rightarrow \mathbb{R}[/itex] on the whole of phi(V) for any chart (U,[itex]\phi [/itex]) containing p, where V is an open neighbourhood of p contained in U.

Is this customary? Why not simply ask that [itex]f\circ \phi^{-1}:\phi(U) \rightarrow \mathbb{R}[/itex] be differentiable at [itex]\phi(p)[/itex]??
 

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quasar987
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I have another question. My text also says that...

If [itex](M,\tau)[/itex] and [itex](N,\nu)[/itex] are manifolds with dim(M)<dim(M), a map [itex]\Phi:M\rightarrow N[/itex] is said to be an immersion if it is locally homeomorphic to it's image.

I assume this means that for all p in M, there exists and open neighbourhood V of p such that the restriction [itex]\Phi |_V:V\rightarrow \Phi(V)[/itex] is a homeomorphism.

My question is: is it assumed that the topologies on V and [itex]\Phi(V)[/itex] used for the notion of continuity are the topologies induced by [itex]\tau[/itex] and [itex]\nu[/itex] respectively? Or are we to use [itex]\tau[/itex] and [itex]\nu[/itex] themselves?
 
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My text defines differentiability of [itex]f:M\rightarrow \mathbb{R}[/itex] at a point p on a manifold M as the differentiability of [itex]f\circ \phi^{-1}:\phi(V) \rightarrow \mathbb{R}[/itex] on the whole of phi(V) for any chart (U,[itex]\phi [/itex]) containing p, where V is an open neighbourhood of p contained in U.

Is this customary? Why not simply ask that [itex]f\circ \phi^{-1}:\phi(U) \rightarrow \mathbb{R}[/itex] be differentiable at [itex]\phi(p)[/itex]??
This is indeed customary. We want a definition that is not dependent on any particular atlas. Furthermore, differentiability is a local property. In determining whether a function is differentiable at a particular point p, we are only concerned with the points near p. The nbd. U as given by the atlas may be very large wrt p and the function f may do weird things away from p even though f is differentiable at p.

A good example to keep in mind is the absolute value function. If we take M=U=R^1 (phi is the identity map), then the definition in your text is as it should be: the absolute value function is indeed differentiable at all non-zero points of M (for every nonzero p in R, take V to be some open interval (p-e,p+e) not containing 0).

(Edited: I snipped an incorrect statement here)

Clearly in higher dimensions you can imagine similarly piecewise linear (not-everywhere-differentiable) functions.
 
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quasar987
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This is indeed customary. We want a definition that is not dependent on any particular atlas. Furthermore, differentiability is a local property. In determining whether a function is differentiable at a particular point p, we are only concerned with the points near p. The nbd. U as given by the atlas may be very large wrt p and the function f may do weird things away from p even though f is differentiable at p.

A good example to keep in mind is the absolute value function. If we take M=U=R^1 (phi is the identity map), then the definition in your text is as it should be: the absolute value function is indeed differentiable at all non-zero points of M (for every nonzero p in R, take V to be some open interval (p-e,p+e) not containing 0).

(Edited: I snipped an incorrect statement here)

Clearly in higher dimensions you can imagine similarly piecewise linear (not-everywhere-differentiable) functions.
Also, I find dubious the definition of differentiability of a function btw two manifolds given in my book. I would rephrase it this way:

Consider M, N two manifolds and a map f:M-->N between them. We say that f is differentiable at a point x of M if for all charts (U,P) of M containing x and all charts (U',P') of N containing f(x), there exists an open neighbourhood [itex]V\subset U[/itex] of x such that [itex]f(V)\subset U'[/itex] and such that the map [itex]P'\circ f\circ P^{-1}:P(V)\rightarrow P'\circ f(V)[/itex] is differentiable on P(V).

We say that f is differentiable on [itex]O\subset M[/itex] if it is differentiable at all points of O.


Is this correct?
 
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Also, I find dubious the definition of differentiability of a function btw two manifolds given in my book. I would rephrase it this way:

Consider M, N two manifolds and a map f:M-->N between them. We say that f is differentiable at a point x of M if for all charts (U,P) of M containing x and all charts (U',P') of N containing f(x), there exists an open neighbourhood [itex]V\subset U[/itex] of x such that [itex]f(V)\subset U'[/itex] and such that the map [itex]P'\circ f\circ P^{-1}:P(V)\rightarrow P'\circ f(V)[/itex] is differentiable on P(V).

We say that f is differentiable on [itex]O\subset M[/itex] if it is differentiable at all points of O.


Is this correct?
I'm not finding anything unkosher about that. It seems like standard boiler-plate... You're going to get the same essential definition of differentiability when dealing with just mappings from R^n to R^m, i.e. in the field of calculus of several variables.

If I were the author, I'd probably change "... all charts..." to "...any chart..." within the text just for flow of reading, but otherwise I'm curious as to your objections.

Actually, take that back: that last definition really should go in the front: first, you define differentiability of the mapping at a point on M, then define differentiability of the mapping on a subset of M.
 
  • #6
quasar987
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I'm not finding anything unkosher about that. It seems like standard boiler-plate... You're going to get the same essential definition of differentiability when dealing with just mappings from R^n to R^m, i.e. in the field of calculus of several variables.

If I were the author, I'd probably change "... all charts..." to "...any chart..." within the text just for flow of reading, but otherwise I'm curious as to your objections.
I have no objections with this definition since it is I who "invented" it.

Actually, take that back: that last definition really should go in the front: first, you define differentiability of the mapping at a point on M, then define differentiability of the mapping on a subset of M.
Isn't this how my definition is constructed? First differentiability at a point of M, then differentiability on a subset of M. Are you suggesting I invert this order?!
 
  • #7
Hurkyl
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In general, things defined WRT open sets behave particularly nicely. Thus, local differentiability is much more preferable to discuss than pointwise differentiability.
 
  • #8
quasar987
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I don't see how differentiability on a set can preceed differentiability at a point, since the former is defined through the later.
 
  • #9
mathwonk
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ask yourself whether there is any significant difference in practice between the two.

i.e. is this a vacuous discussion?
 
  • #10
quasar987
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Are you answering to post #2 wonk?
 
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I have no objections with this definition since it is I who "invented" it.
sorry. i misread your post. it is perfectly kosher.
 
  • #12
mathwonk
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actually that was not the real me who posted that smart alec remark, but the bizarro mathwonk who inhabits my body at odd times, especially at night.
 
  • #13
quasar987
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And what does the authentic wonk has to say?
 
  • #14
mathwonk
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as to post #2, your text is a little unclear, as an immersion should be a differentiable map which ahs those other properties, i.e. every point p of the source should have a nbhd such that the restriction to that nbhd is an embedding onto a submanifold of the target.

no wait a minute actually your book is wrong, as a map like the one taking R to R^2 by sending t to (t^2,t^3) is a homeomorphism globally onto its image in the induced topology but is not an imersion sinceb the derivative is zero at t=0. so semthing is fishy with your books definition I personally think.

of course some of these etrms are up to personal taste as to how tod efine the but i submit that no self respecting differential geometer would agree that my map above is an immersion.

see the problem is that the image set of my map is not a diferentiable submanifold, but your definition did not require that did it?
 
  • #15
mathwonk
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actually the question as asked i post #2 is hard for me to understand as i cannot see any way to use the topology of the target except as the induced topology on the image.
 
  • #16
mathwonk
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actually the bizarro mathwonk may have taken over my body completely.
 

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