Question on differentiable manifolds and tangent spaces

[mnb96]

Hello,

I notice that most books on differential geometry introduce the definition of differentiable manifold by describing what I would regard as a differentiable manifold of class C^∞ (i.e. a smooth manifold).

Why so?
Don't we simply need a class C¹ differentiable manifold in order to have tangent spaces and do differential geometry?

What do we need the partial derivatives of all orders, in particular of third, fourth, fifth order for?
The Jacobian is made of only first-order partial derivatives after all.

 

[jgens]

There is a theorem that says every C^k where k ≥ 1 has a compatible smooth structure. There are also some theorems to the effect that if you are studying things like whether one manifold immerses into another, then the answer to this question is the same in the C^k setting and in the smooth setting. So assuming you only care about things like immersions and diffeomorphisms of manifolds, putting these two guys together says we lose no generality by restricting our attention to the smooth case. In some situations, particularly in analysis, you do need to pay attention to differentiability type so restricting to smooth manifolds is not always possible.

Why you might want higher differentiability type depends on what exactly you are doing with your manifold. Anything dealing with Morse theory is going to require your manifold to be at least C² and some results in that arena require still higher differentiability type.

 

[mnb96]

Thanks jgens,

could you explain what do you mean in this context by "having compatible smooth structure" ?
I am afraid I am not familiar with this concept.

 

[jgens]

Let (M,A) be a C¹ manifold where A is the maximal atlas of C¹ charts. If (M,B) is a smooth manifold, where B is a maximal atlas of smooth charts, then this smooth structure is compatible with the C¹ structure if and only if the maximal C¹ atlas generated by B is exactly A.

 

[lavinia]

Differential geometry studies the idea of curvature which in all cases that I have seen requires at least 2 derivatives. There may be generalizations of curvature that do not.

Many problems in analysis require more than one derivative since partial differential equations can involve derivatives of any order.The assumption of smoothness removes the headache of worrying about the degree of differentiability of a coordinate transformation.