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an important hypothesis was missing from the lemma.
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alvarezpaiva
alvarezpaiva
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This should be a little easier than using Sard's theorem:

The manifold $H$ of hyperplanes in $(n+1)$-space has dimension $n+1$. The map from $M$ to $H$ that sends each point to its tangent hyperplane is a differentiable map from an $n$-dimensional manifold to an $(n+1)$-dimensional manifold. The image must have measure zero.

You just need a lemma saying that if $f: U \rightarrow \mathbb{R}^m$ is a $C^1$ map from an open subset of $\mathbb{R}^k$ to $\mathbb{R}^m$ with $k < m$, then the image has measure zero. This is also a lemma for the Morse-Sard theorem (see Hirsch's Differential Topology).

This should be a little easier than using Sard's theorem:

The manifold $H$ of hyperplanes in $(n+1)$-space has dimension $n+1$. The map from $M$ to $H$ that sends each point to its tangent hyperplane is a differentiable map from an $n$-dimensional manifold to an $(n+1)$-dimensional manifold. The image must have measure zero.

You just need a lemma saying that if $f: U \rightarrow \mathbb{R}^m$ is a $C^1$ map from an open subset of $\mathbb{R}^k$ to $\mathbb{R}^m$, then the image has measure zero. This is also a lemma for the Morse-Sard theorem (see Hirsch's Differential Topology).

This should be a little easier than using Sard's theorem:

The manifold $H$ of hyperplanes in $(n+1)$-space has dimension $n+1$. The map from $M$ to $H$ that sends each point to its tangent hyperplane is a differentiable map from an $n$-dimensional manifold to an $(n+1)$-dimensional manifold. The image must have measure zero.

You just need a lemma saying that if $f: U \rightarrow \mathbb{R}^m$ is a $C^1$ map from an open subset of $\mathbb{R}^k$ to $\mathbb{R}^m$ with $k < m$, then the image has measure zero. This is also a lemma for the Morse-Sard theorem (see Hirsch's Differential Topology).

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Ben McKay
Ben McKay
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This souldshould be a little easier than using Sard's theorem:

The manifold $H$ of hyperplanes in $(n+1)$-space has dimension $n+1$, the. The map from $M$ to $H$ that sends each point to its tangent hyperplane is a differentiable map from an $n$-dimensional manifold to an $(n+1)$-dimensional manifold. The image must have measure zero.

You just need a lemma saying that if $f: U \rightarrow \mathbb{R}^m$ is a $C^1$ map from an open subset of $\mathbb{R}^k$ to $\mathbb{R}^m$, then the image has measure zero. This is also a lemma for the Morse-Sard theorem (see Hirsch's Differential Topology).

This sould be a little easier than using Sard's theorem:

The manifold $H$ of hyperplanes in $(n+1)$-space has dimension $n+1$, the map from $M$ to $H$ that sends each point to its tangent hyperplane is a differentiable map from an $n$-dimensional manifold to an $(n+1)$-dimensional manifold. The image must have measure zero.

You just need a lemma saying that if $f: U \rightarrow \mathbb{R}^m$ is a $C^1$ map from an open subset of $\mathbb{R}^k$ to $\mathbb{R}^m$, then the image has measure zero. This is also a lemma for the Morse-Sard theorem (see Hirsch's Differential Topology).

This should be a little easier than using Sard's theorem:

The manifold $H$ of hyperplanes in $(n+1)$-space has dimension $n+1$. The map from $M$ to $H$ that sends each point to its tangent hyperplane is a differentiable map from an $n$-dimensional manifold to an $(n+1)$-dimensional manifold. The image must have measure zero.

You just need a lemma saying that if $f: U \rightarrow \mathbb{R}^m$ is a $C^1$ map from an open subset of $\mathbb{R}^k$ to $\mathbb{R}^m$, then the image has measure zero. This is also a lemma for the Morse-Sard theorem (see Hirsch's Differential Topology).

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alvarezpaiva
alvarezpaiva
  • 13.5k
  • 40
  • 83

This sould be a little easier than using Sard's theorem:

The manifold $H$ of hyperplanes in $(n+1)$-space has dimension $n+1$, the map from $M$ to $H$ that sends each point to its tangent hyperplane is a differentiable map from an $n$-dimensional manifold to an $(n+1)$-dimensional manifold. The image must have measure zero.

You just need a lemma saying that if $f: U \rightarrow \mathbb{R}^m$ is a $C^1$ map from an open subset of $\mathbb{R}^k$ to $\mathbb{R}^m$, then the image has measure zero. This is also a lemma for the Morse-Sard theorem (see Hirsch's Differential Topology).