Bounding the volume distortion of a manifold

In summary, when considering a compact set $U$ in $\mathbb{R}^k$ and a $C^1$ bijection $f:U\to\mathbb{R}^n$, the volume distortion of the resulting manifold $M=f(U)$ is defined as $G=det(DftDf)$. For $n=1$, it can be shown that $G=1+|\nabla f|^2$. However, for $n>1$, it is possible to bound $G$ from below under certain assumptions, such as taking a finite sub-cover and considering $g_i:\mathbb{R^{n-1}\to R}$ as a $C^1$ function
  • #1
eyenir
3
0
Let $U$ be a compact set in $\mathbb{R}^k$ and let $f:U\to\mathbb{R}^n$ be a $C^1$ bijection. Consider the manifold $M=f(U)$.

Its volume distortion is defined as $G=det(DftDf).$ If $n=1$, one can deduce that $G=1+|\nabla f|^2$.

What happens for $n>1$? Can one bound from below this $G$? If so: under which assumptions?
 
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  • #2
eyenir said:
Let $U$ be a compact set in $\mathbb{R}^k$ and let $f:U\to\mathbb{R}^n$ be a $C^1$ bijection.
did not you know that the image of a compact set under a continuous mapping is a compact set?
 
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  • #3
So you suggest taking a finite sub-cover ${U_i}$ such that $M|_{U_i}=\{(x_1,\dots,x_{n-1},g_i\}$ where $g_i:\mathbb{R^{n-1}\to R}$ is a $C^1$ function on $U_i$?
 

FAQ: Bounding the volume distortion of a manifold

1. What is volume distortion in relation to manifolds?

Volume distortion refers to the change in volume that occurs when a manifold is transformed from one coordinate system to another. This distortion can be measured by calculating the determinant of the Jacobian matrix, which represents the change in scale at each point on the manifold.

2. Why is it important to bound the volume distortion of a manifold?

Bounding the volume distortion allows us to understand the amount of distortion that occurs when transforming between coordinate systems on a manifold. This is crucial for accurately representing data and making meaningful comparisons between different manifolds.

3. How is the volume distortion of a manifold calculated?

The volume distortion of a manifold can be calculated by finding the determinant of the Jacobian matrix, which represents the change in scale at each point on the manifold. This can be done using mathematical formulas or by using specialized software programs.

4. What factors can affect the volume distortion of a manifold?

There are several factors that can affect the volume distortion of a manifold, including the curvature of the manifold, the choice of coordinate system, and the complexity of the manifold itself. In general, the more curved and complex a manifold is, the higher the volume distortion will be.

5. Can the volume distortion of a manifold be completely eliminated?

No, it is not possible to completely eliminate volume distortion when transforming between coordinate systems on a manifold. This is because any transformation will inevitably change the scale of the manifold at some points. However, by bounding the volume distortion, we can minimize its impact and ensure more accurate representations of the data on the manifold.

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