Mixing
On morse theory and foliations
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Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: what can one say about the critical points, i.e. $t^{-1}(c)$ where $c$ is the critical point. Looking at examples, I see why this fails to be even a smooth submanifold. This raises the question, as to what we can say about it.
My intuition leads me to ask the following questions: Is it some sort of a stratification? If yes, is it well studied, and where can I find it? Also, if the ambient manifold has a metric structure, is it possible to use it in anyway to induce anything on the substructure?
differential-geometry riemannian-geometry morse-theory semi-riemannian-geometry foliations
asked Jan 18 at 17:09
Sandesh JrSandesh Jr
867
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add a comment |
$begingroup$
Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: what can one say about the critical points, i.e. $t^{-1}(c)$ where $c$ is the critical point. Looking at examples, I see why this fails to be even a smooth submanifold. This raises the question, as to what we can say about it.
My intuition leads me to ask the following questions: Is it some sort of a stratification? If yes, is it well studied, and where can I find it? Also, if the ambient manifold has a metric structure, is it possible to use it in anyway to induce anything on the substructure?
differential-geometry riemannian-geometry morse-theory semi-riemannian-geometry foliations
asked Jan 18 at 17:09
Sandesh JrSandesh Jr
867
$endgroup$
$begingroup$
Morse theory says that critical sets are either points (for the local max and min) of singular manifolds with isolated quadratic singularities which are described in terms of the Morse index. This will be discussed in any book on Morse theory. What else would you want to know? As for a metric (Riemannian or Lorentzian), sure, you can restrict it to the critical sets away from the singularities, but the restriction could be degenerate (in the Lorentzian case).
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– Moishe Cohen
Jan 18 at 18:05
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I see. So the critical sets are singular manifolds? Is there a good reference which talks about singular manifolds?
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– Sandesh Jr
Jan 19 at 10:41
$begingroup$
See my answer here: math.stackexchange.com/questions/848448/…
$endgroup$
– Moishe Cohen
Jan 29 at 16:12
add a comment |
$begingroup$
Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: what can one say about the critical points, i.e. $t^{-1}(c)$ where $c$ is the critical point. Looking at examples, I see why this fails to be even a smooth submanifold. This raises the question, as to what we can say about it.
My intuition leads me to ask the following questions: Is it some sort of a stratification? If yes, is it well studied, and where can I find it? Also, if the ambient manifold has a metric structure, is it possible to use it in anyway to induce anything on the substructure?
differential-geometry riemannian-geometry morse-theory semi-riemannian-geometry foliations
asked Jan 18 at 17:09
Sandesh JrSandesh Jr
867
$endgroup$
Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: what can one say about the critical points, i.e. $t^{-1}(c)$ where $c$ is the critical point. Looking at examples, I see why this fails to be even a smooth submanifold. This raises the question, as to what we can say about it.
My intuition leads me to ask the following questions: Is it some sort of a stratification? If yes, is it well studied, and where can I find it? Also, if the ambient manifold has a metric structure, is it possible to use it in anyway to induce anything on the substructure?
differential-geometry riemannian-geometry morse-theory semi-riemannian-geometry foliations
differential-geometry riemannian-geometry morse-theory semi-riemannian-geometry foliations
asked Jan 18 at 17:09
Sandesh JrSandesh Jr
867
asked Jan 18 at 17:09
Sandesh JrSandesh Jr
867
asked Jan 18 at 17:09
Sandesh JrSandesh Jr
867
asked Jan 18 at 17:09
Sandesh JrSandesh Jr
867
asked Jan 18 at 17:09
Sandesh JrSandesh Jr
867
867
$begingroup$
Morse theory says that critical sets are either points (for the local max and min) of singular manifolds with isolated quadratic singularities which are described in terms of the Morse index. This will be discussed in any book on Morse theory. What else would you want to know? As for a metric (Riemannian or Lorentzian), sure, you can restrict it to the critical sets away from the singularities, but the restriction could be degenerate (in the Lorentzian case).
$endgroup$
– Moishe Cohen
Jan 18 at 18:05
$begingroup$
I see. So the critical sets are singular manifolds? Is there a good reference which talks about singular manifolds?
$endgroup$
– Sandesh Jr
Jan 19 at 10:41
$begingroup$
See my answer here: math.stackexchange.com/questions/848448/…
$endgroup$
– Moishe Cohen
Jan 29 at 16:12
add a comment |
$begingroup$
Morse theory says that critical sets are either points (for the local max and min) of singular manifolds with isolated quadratic singularities which are described in terms of the Morse index. This will be discussed in any book on Morse theory. What else would you want to know? As for a metric (Riemannian or Lorentzian), sure, you can restrict it to the critical sets away from the singularities, but the restriction could be degenerate (in the Lorentzian case).
$endgroup$
– Moishe Cohen
Jan 18 at 18:05
$begingroup$
I see. So the critical sets are singular manifolds? Is there a good reference which talks about singular manifolds?
$endgroup$
– Sandesh Jr
Jan 19 at 10:41
$begingroup$
See my answer here: math.stackexchange.com/questions/848448/…
$endgroup$
– Moishe Cohen
Jan 29 at 16:12
$begingroup$
Morse theory says that critical sets are either points (for the local max and min) of singular manifolds with isolated quadratic singularities which are described in terms of the Morse index. This will be discussed in any book on Morse theory. What else would you want to know? As for a metric (Riemannian or Lorentzian), sure, you can restrict it to the critical sets away from the singularities, but the restriction could be degenerate (in the Lorentzian case).
$endgroup$
– Moishe Cohen
Jan 18 at 18:05
$begingroup$
Morse theory says that critical sets are either points (for the local max and min) of singular manifolds with isolated quadratic singularities which are described in terms of the Morse index. This will be discussed in any book on Morse theory. What else would you want to know? As for a metric (Riemannian or Lorentzian), sure, you can restrict it to the critical sets away from the singularities, but the restriction could be degenerate (in the Lorentzian case).
$endgroup$
– Moishe Cohen
Jan 18 at 18:05
$begingroup$
I see. So the critical sets are singular manifolds? Is there a good reference which talks about singular manifolds?
$endgroup$
– Sandesh Jr
Jan 19 at 10:41
$begingroup$
I see. So the critical sets are singular manifolds? Is there a good reference which talks about singular manifolds?
$endgroup$
– Sandesh Jr
Jan 19 at 10:41
$begingroup$
See my answer here: math.stackexchange.com/questions/848448/…
$endgroup$
– Moishe Cohen
Jan 29 at 16:12
$begingroup$
See my answer here: math.stackexchange.com/questions/848448/…
$endgroup$
– Moishe Cohen
Jan 29 at 16:12
add a comment |
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$begingroup$
Morse theory says that critical sets are either points (for the local max and min) of singular manifolds with isolated quadratic singularities which are described in terms of the Morse index. This will be discussed in any book on Morse theory. What else would you want to know? As for a metric (Riemannian or Lorentzian), sure, you can restrict it to the critical sets away from the singularities, but the restriction could be degenerate (in the Lorentzian case).
$endgroup$
– Moishe Cohen
Jan 18 at 18:05
$begingroup$
I see. So the critical sets are singular manifolds? Is there a good reference which talks about singular manifolds?
$endgroup$
– Sandesh Jr
Jan 19 at 10:41
$begingroup$
See my answer here: math.stackexchange.com/questions/848448/…
$endgroup$
– Moishe Cohen
Jan 29 at 16:12