Question on differentiable manifolds and tangent spaces

In summary, the conversation discusses the differentiability types required for studying differential geometry and how the smooth case is often sufficient for various purposes. The concept of "compatible smooth structure" is also mentioned as well as the importance of higher differentiability types in certain situations.
  • #1
mnb96
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5
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.
 
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  • #2
There is a theorem that says every Ck 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 Ck 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.
 
  • #3
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.
 
  • #4
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.
 
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  • #5
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.
 
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1. What is a differentiable manifold?

A differentiable manifold is a mathematical concept that describes a space that is locally similar to Euclidean space and can be smoothly mapped to it. It is a generalization of the notion of a curve or surface in traditional geometry. It is used in many areas of mathematics, including differential geometry, topology, and physics.

2. What is the difference between a differentiable manifold and a smooth manifold?

There is no difference between a differentiable manifold and a smooth manifold. They are both terms used to describe a manifold that can be smoothly mapped to Euclidean space. The choice of term may vary depending on the context or the preferences of the mathematician.

3. What is a tangent space?

A tangent space is a mathematical concept that describes the local behavior of a differentiable manifold at a specific point. It is a vector space that is tangent to the manifold at that point and contains all the possible directions in which the manifold can be smoothly extended.

4. How is the tangent space related to the concept of a derivative?

In the context of differentiable manifolds, the tangent space is closely related to the derivative. The derivative of a function on a manifold is a linear map from the tangent space at one point to the tangent space at another point. This allows us to generalize the concept of a derivative from traditional calculus to higher-dimensional spaces.

5. What are some real-world applications of differentiable manifolds and tangent spaces?

Differentiable manifolds and tangent spaces have several real-world applications, including computer graphics, robotics, and machine learning. They are also used in physics to describe the behavior of objects in space-time and in economics to model complex systems. These concepts are also fundamental to the understanding of differential geometry and topology, which have numerous applications in mathematics and other fields.

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