Differentiability of a function between manifolds

In summary, two submanifolds $$M^n \subset \mathbb{R}^N$$ and $$N^k \subset \mathbb{R}^K$$ are given. A function $$f : M \rightarrow N$$ is differentiable if for every map $$(U,\varphi)$$ of M, the transformation $$f \circ \varphi^{-1}$$ is differentiable. This means that it is differentiable in the sense of a "normal" function in several variables, and can be expressed as $$d(f \circ \varphi^{-1})(x)=df(\varphi^{-1}(x))d\varphi^{-1}(x)$$. This definition can be further refined
  • #1
Maxi1995
14
0
Hello,
let $$M^n \subset \mathbb{R}^N$$ $$N^k \subset \mathbb{R}^K$$
be two submanifolds.
We say a function $$f : M \rightarrow N$$ is differentiable if and only if for every map $$(U,\varphi)$$ of M the transformation

$$f \circ \varphi^{-1}: \varphi(U) \subset \mathbb{R}^N \rightarrow \mathbb{R}^K$$

is differentiable.

Does this mean that it is differentiable in the sense of a "normal" function in several variables, thus to say $$d(f \circ \varphi^{-1})(x)=df(\varphi^{-1}(x))d\varphi^{-1}(x)$$?
 
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  • #2
You do not have ##df(\ldots)## while are defining it. You transform ##f## to ##g=f\circ \varphi^{-1}## which is a real valued function which you know how to differentiate. Thus you have ##dg=df\,d\varphi^{-1}## which defines you ##df##.
 
  • #3
Thank you very much. :bow:
 

1. What is a manifold and how does it relate to functions?

A manifold is a mathematical concept that describes a space that is locally similar to Euclidean space. It is a generalization of the concept of a surface in three-dimensional space. A function between manifolds is a mapping that assigns each point on one manifold to a point on another manifold, similar to how a regular function assigns each input value to an output value.

2. How is differentiability defined for a function between manifolds?

Differentiability for a function between manifolds is defined in terms of tangent spaces. A function is differentiable between manifolds if, at each point, it has a well-defined linear transformation between the tangent spaces of the two manifolds. This means that the function is locally smooth and can be approximated by a linear function between the two manifolds.

3. What is the importance of differentiability for functions between manifolds?

Differentiability is important because it allows us to study the behavior of a function between manifolds in a local sense. It also enables us to define important concepts such as derivatives and gradients, which are essential for understanding the behavior of a function in a given direction and for solving optimization problems.

4. How is differentiability related to continuity for functions between manifolds?

In general, a function between manifolds can be differentiable without being continuous. However, if a function is differentiable, it must also be continuous. This means that a function between manifolds can be considered to be a stronger condition than continuity, as it requires the function to have a well-defined linear transformation at each point.

5. Can a function be differentiable at some points but not others between manifolds?

Yes, a function can be differentiable at some points but not others between manifolds. This is because differentiability is defined at each point individually, and a function may have a well-defined linear transformation at one point but not at another. However, if a function is differentiable at all points between two manifolds, it is said to be differentiable as a whole.

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