Is this a submanifold?












3












$begingroup$


Let $(M,g)$ be a compact Riemannian manifold with an isometric action $rho : G to mathrm{Iso}(M)$ by a compact Lie group $G$. There is a natural extension of $rho$ to $TM$ given by:
$$psi : G times TM to TM$$
$$psi(g,p,X) := (rho(g)p,drho(g)X).$$



I would like to know if the set:
$$TM supset tilde S := {(p,X) in TM : G_X = G_p}$$ is a submanifold of $TM$, and further, if $pi : TM to M$ is the natural projection, then $pi(tilde S)$ is a submanifold of $M$.



I tried the following approach:



For each $gin G$ consider $eta_g(p,X) equiv eta(g,p,X) := d^2_{TM}left((p,X),psi(g,p,X)right).$ Then, one has:
$$eta_g^{-1}(0) = {(p,X) :(rho(g)p,drho(g)X) = (p,X)}.$$
So,
$$tilde S = bigcup_{gin G}eta_g^{-1}(0). $$



But according to my calculation $0$ is not a regular value of $eta_g$.



I appreciate any help.



EDIT



Thanks to all the answers and comments. I shall change a little the candidate to manifold to another that will be more helpful to me.



I would like to know if one denotes by $cal H_p$ the orthogonal complement to $T_pGcdot p$ on the $g$-metric, then the set
$$S := {p in M : exists X in mathcal H_p : G_X = G_p}$$
is a submanifold of $M$. In fact, this is what I was trying to prove at first.










share|cite|improve this question











$endgroup$












  • $begingroup$
    Is this a pigeon? ;-)
    $endgroup$
    – David Roberts
    Mar 9 at 21:54










  • $begingroup$
    @DavidRoberts, I am sorry, what are you asking if it is a pigeon?
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 22:11






  • 4




    $begingroup$
    It's a meme, meant in good humour.
    $endgroup$
    – David Roberts
    Mar 9 at 22:57










  • $begingroup$
    By $G_X$ and $G_p$ you are referring to the vector and point stabilizers, respectively? I would imagine that generally this is not a manifold without some pretty restrictive assumptions. I'm imagining the action of $G$ degenerating along a stratum, with the stabilized vector space changing. What are you hoping to do with a result like this?
    $endgroup$
    – Ryan Budney
    Mar 10 at 0:41












  • $begingroup$
    Dear @RyanBudney, please, consider seeing my edit, as It was suggested on the comments $S$ possibly is a manifold, and that was my original problem. I am working on positive curvature and I need on some step of my argument to be sure that $S$ is a submanifold.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 0:43
















3












$begingroup$


Let $(M,g)$ be a compact Riemannian manifold with an isometric action $rho : G to mathrm{Iso}(M)$ by a compact Lie group $G$. There is a natural extension of $rho$ to $TM$ given by:
$$psi : G times TM to TM$$
$$psi(g,p,X) := (rho(g)p,drho(g)X).$$



I would like to know if the set:
$$TM supset tilde S := {(p,X) in TM : G_X = G_p}$$ is a submanifold of $TM$, and further, if $pi : TM to M$ is the natural projection, then $pi(tilde S)$ is a submanifold of $M$.



I tried the following approach:



For each $gin G$ consider $eta_g(p,X) equiv eta(g,p,X) := d^2_{TM}left((p,X),psi(g,p,X)right).$ Then, one has:
$$eta_g^{-1}(0) = {(p,X) :(rho(g)p,drho(g)X) = (p,X)}.$$
So,
$$tilde S = bigcup_{gin G}eta_g^{-1}(0). $$



But according to my calculation $0$ is not a regular value of $eta_g$.



I appreciate any help.



EDIT



Thanks to all the answers and comments. I shall change a little the candidate to manifold to another that will be more helpful to me.



I would like to know if one denotes by $cal H_p$ the orthogonal complement to $T_pGcdot p$ on the $g$-metric, then the set
$$S := {p in M : exists X in mathcal H_p : G_X = G_p}$$
is a submanifold of $M$. In fact, this is what I was trying to prove at first.










share|cite|improve this question











$endgroup$












  • $begingroup$
    Is this a pigeon? ;-)
    $endgroup$
    – David Roberts
    Mar 9 at 21:54










  • $begingroup$
    @DavidRoberts, I am sorry, what are you asking if it is a pigeon?
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 22:11






  • 4




    $begingroup$
    It's a meme, meant in good humour.
    $endgroup$
    – David Roberts
    Mar 9 at 22:57










  • $begingroup$
    By $G_X$ and $G_p$ you are referring to the vector and point stabilizers, respectively? I would imagine that generally this is not a manifold without some pretty restrictive assumptions. I'm imagining the action of $G$ degenerating along a stratum, with the stabilized vector space changing. What are you hoping to do with a result like this?
    $endgroup$
    – Ryan Budney
    Mar 10 at 0:41












  • $begingroup$
    Dear @RyanBudney, please, consider seeing my edit, as It was suggested on the comments $S$ possibly is a manifold, and that was my original problem. I am working on positive curvature and I need on some step of my argument to be sure that $S$ is a submanifold.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 0:43














3












3








3


1



$begingroup$


Let $(M,g)$ be a compact Riemannian manifold with an isometric action $rho : G to mathrm{Iso}(M)$ by a compact Lie group $G$. There is a natural extension of $rho$ to $TM$ given by:
$$psi : G times TM to TM$$
$$psi(g,p,X) := (rho(g)p,drho(g)X).$$



I would like to know if the set:
$$TM supset tilde S := {(p,X) in TM : G_X = G_p}$$ is a submanifold of $TM$, and further, if $pi : TM to M$ is the natural projection, then $pi(tilde S)$ is a submanifold of $M$.



I tried the following approach:



For each $gin G$ consider $eta_g(p,X) equiv eta(g,p,X) := d^2_{TM}left((p,X),psi(g,p,X)right).$ Then, one has:
$$eta_g^{-1}(0) = {(p,X) :(rho(g)p,drho(g)X) = (p,X)}.$$
So,
$$tilde S = bigcup_{gin G}eta_g^{-1}(0). $$



But according to my calculation $0$ is not a regular value of $eta_g$.



I appreciate any help.



EDIT



Thanks to all the answers and comments. I shall change a little the candidate to manifold to another that will be more helpful to me.



I would like to know if one denotes by $cal H_p$ the orthogonal complement to $T_pGcdot p$ on the $g$-metric, then the set
$$S := {p in M : exists X in mathcal H_p : G_X = G_p}$$
is a submanifold of $M$. In fact, this is what I was trying to prove at first.










share|cite|improve this question











$endgroup$




Let $(M,g)$ be a compact Riemannian manifold with an isometric action $rho : G to mathrm{Iso}(M)$ by a compact Lie group $G$. There is a natural extension of $rho$ to $TM$ given by:
$$psi : G times TM to TM$$
$$psi(g,p,X) := (rho(g)p,drho(g)X).$$



I would like to know if the set:
$$TM supset tilde S := {(p,X) in TM : G_X = G_p}$$ is a submanifold of $TM$, and further, if $pi : TM to M$ is the natural projection, then $pi(tilde S)$ is a submanifold of $M$.



I tried the following approach:



For each $gin G$ consider $eta_g(p,X) equiv eta(g,p,X) := d^2_{TM}left((p,X),psi(g,p,X)right).$ Then, one has:
$$eta_g^{-1}(0) = {(p,X) :(rho(g)p,drho(g)X) = (p,X)}.$$
So,
$$tilde S = bigcup_{gin G}eta_g^{-1}(0). $$



But according to my calculation $0$ is not a regular value of $eta_g$.



I appreciate any help.



EDIT



Thanks to all the answers and comments. I shall change a little the candidate to manifold to another that will be more helpful to me.



I would like to know if one denotes by $cal H_p$ the orthogonal complement to $T_pGcdot p$ on the $g$-metric, then the set
$$S := {p in M : exists X in mathcal H_p : G_X = G_p}$$
is a submanifold of $M$. In fact, this is what I was trying to prove at first.







dg.differential-geometry riemannian-geometry differential-topology group-actions






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 10 at 0:42







L.F. Cavenaghi

















asked Mar 9 at 18:35









L.F. CavenaghiL.F. Cavenaghi

610213




610213












  • $begingroup$
    Is this a pigeon? ;-)
    $endgroup$
    – David Roberts
    Mar 9 at 21:54










  • $begingroup$
    @DavidRoberts, I am sorry, what are you asking if it is a pigeon?
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 22:11






  • 4




    $begingroup$
    It's a meme, meant in good humour.
    $endgroup$
    – David Roberts
    Mar 9 at 22:57










  • $begingroup$
    By $G_X$ and $G_p$ you are referring to the vector and point stabilizers, respectively? I would imagine that generally this is not a manifold without some pretty restrictive assumptions. I'm imagining the action of $G$ degenerating along a stratum, with the stabilized vector space changing. What are you hoping to do with a result like this?
    $endgroup$
    – Ryan Budney
    Mar 10 at 0:41












  • $begingroup$
    Dear @RyanBudney, please, consider seeing my edit, as It was suggested on the comments $S$ possibly is a manifold, and that was my original problem. I am working on positive curvature and I need on some step of my argument to be sure that $S$ is a submanifold.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 0:43


















  • $begingroup$
    Is this a pigeon? ;-)
    $endgroup$
    – David Roberts
    Mar 9 at 21:54










  • $begingroup$
    @DavidRoberts, I am sorry, what are you asking if it is a pigeon?
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 22:11






  • 4




    $begingroup$
    It's a meme, meant in good humour.
    $endgroup$
    – David Roberts
    Mar 9 at 22:57










  • $begingroup$
    By $G_X$ and $G_p$ you are referring to the vector and point stabilizers, respectively? I would imagine that generally this is not a manifold without some pretty restrictive assumptions. I'm imagining the action of $G$ degenerating along a stratum, with the stabilized vector space changing. What are you hoping to do with a result like this?
    $endgroup$
    – Ryan Budney
    Mar 10 at 0:41












  • $begingroup$
    Dear @RyanBudney, please, consider seeing my edit, as It was suggested on the comments $S$ possibly is a manifold, and that was my original problem. I am working on positive curvature and I need on some step of my argument to be sure that $S$ is a submanifold.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 0:43
















$begingroup$
Is this a pigeon? ;-)
$endgroup$
– David Roberts
Mar 9 at 21:54




$begingroup$
Is this a pigeon? ;-)
$endgroup$
– David Roberts
Mar 9 at 21:54












$begingroup$
@DavidRoberts, I am sorry, what are you asking if it is a pigeon?
$endgroup$
– L.F. Cavenaghi
Mar 9 at 22:11




$begingroup$
@DavidRoberts, I am sorry, what are you asking if it is a pigeon?
$endgroup$
– L.F. Cavenaghi
Mar 9 at 22:11




4




4




$begingroup$
It's a meme, meant in good humour.
$endgroup$
– David Roberts
Mar 9 at 22:57




$begingroup$
It's a meme, meant in good humour.
$endgroup$
– David Roberts
Mar 9 at 22:57












$begingroup$
By $G_X$ and $G_p$ you are referring to the vector and point stabilizers, respectively? I would imagine that generally this is not a manifold without some pretty restrictive assumptions. I'm imagining the action of $G$ degenerating along a stratum, with the stabilized vector space changing. What are you hoping to do with a result like this?
$endgroup$
– Ryan Budney
Mar 10 at 0:41






$begingroup$
By $G_X$ and $G_p$ you are referring to the vector and point stabilizers, respectively? I would imagine that generally this is not a manifold without some pretty restrictive assumptions. I'm imagining the action of $G$ degenerating along a stratum, with the stabilized vector space changing. What are you hoping to do with a result like this?
$endgroup$
– Ryan Budney
Mar 10 at 0:41














$begingroup$
Dear @RyanBudney, please, consider seeing my edit, as It was suggested on the comments $S$ possibly is a manifold, and that was my original problem. I am working on positive curvature and I need on some step of my argument to be sure that $S$ is a submanifold.
$endgroup$
– L.F. Cavenaghi
Mar 10 at 0:43




$begingroup$
Dear @RyanBudney, please, consider seeing my edit, as It was suggested on the comments $S$ possibly is a manifold, and that was my original problem. I am working on positive curvature and I need on some step of my argument to be sure that $S$ is a submanifold.
$endgroup$
– L.F. Cavenaghi
Mar 10 at 0:43










2 Answers
2






active

oldest

votes


















4












$begingroup$

Your definition implies that
$$ tilde S = bigcup_{pin M} T_pM^{G_p}. $$
In particular, $pi(tilde S) = M$, and $tilde S$ will be a submanifold of $TM$ iff the dimension of $T_pM^{G_p}$ is the same for all $pin M$.



$G$ being connected won't necessarily make this happen: Another counterexample is $S^1$ acting by rotation on $S^2$, fixing the north and south poles, $n$ and $s$. $tilde S$ is then $T(S^2setminus{n,s})cup {n,s}$, and the points $n$ and $s$ do not have neighborhoods homeomorphic to open balls.



EDIT



To address the modified question: If $pin S$, then the orbit $Gp$ has a neighborhood homeomorphic to $Gtimes_{G_p} mathcal H_p$ and we're assuming that $mathcal H_p^{G_p} neq 0$. If $qin mathcal H_p$ is any point, then $mathcal H_p^{G_p}$ will be fixed by $G_q$ and still be orthogonal to $Gq$, hence $q$ and every point in $Gq$ will be in $S$. This shows that a neighborhood of $Gp$ is contained in $S$. Thus $S$ is an open submanifold of $M$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    what if we change a little bit my definition by asking the following: $S = {p in M : exists 0 neq X in mathcal {H}_p subset T_pM : G_X = G_p}$? Now I ask if such $S$ is a submanifold of $M$, where $cal H_p$ is the space $g$-orthogonal to $T_pGcdot p$.
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 23:13








  • 1




    $begingroup$
    My first reaction is that, yes, $S$ will be a submanifold. In fact, an open submanifold: If $pin S$ then all points in an open neighborhood of $p$ should also satisfy your new condition. Essentially, a neighborhood of $p$ will look like $T_p M$ with the action of $G_p$.
    $endgroup$
    – Steve Costenoble
    Mar 10 at 0:11










  • $begingroup$
    I have edited the question in order to encompass $S$ to it, it would be extremely helpful if you explain to me your thoughts on why $S$ has a chance to be a submanifold.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 0:45












  • $begingroup$
    I will accept your answer since I could prove it was true following your comment, I will post the proof on the question.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 18:45



















2












$begingroup$

For the obvious reflection action of $mathbb{Z}/2mathbb{Z}$ on $S^1$ this set $tilde S $ has two singular points at the zero vectors tangent at points $p=(1,0)$ and $q=(-1,0)$.
Then $tilde S$ is $T(S^1setminus {p,q}) cup {p_0,q_0}$ where $p_0, q_0$ are zero vectors at $p,q$. This is a connected set and after removing $p_0,q_0$ we obtain a disconnected set. So obviously it is not a manifold because every connected manifold of dimension at least $2$, remains connected after removing a finite set.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    "every connected manifold remains connected after removing a finite set" -- this is not true for 1-dimensional manifolds. Nonetheless, your example is valid: if $x$ is one of the fixed points of the reflection action, then a neighborhood of $(x,0)intilde S$ is homeomorphic to $mathbb R^2$ minus the coordinate axes but including the origin, so it cannot be a manifold.
    $endgroup$
    – Arun Debray
    Mar 9 at 21:42










  • $begingroup$
    @ArunDebray yes thank you but $tilde S $ is 2 dimensional at generic points. I revise the answer. Thanks again for your correction!
    $endgroup$
    – Ali Taghavi
    Mar 9 at 21:44






  • 1




    $begingroup$
    @AliTaghavi, thank you very much. Do you know if one can restrict some hypothesis in order to obtain a manifold? Or, what kind of structure this set has?
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 22:18










  • $begingroup$
    You are well come. What about if we assume $G$ is connected? I am not sure what is the answer in this case.
    $endgroup$
    – Ali Taghavi
    Mar 9 at 22:29






  • 1




    $begingroup$
    @AliTaghavi, take a look to the other answer and possible on the comments. I changed a little the problem.
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 23:15











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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









4












$begingroup$

Your definition implies that
$$ tilde S = bigcup_{pin M} T_pM^{G_p}. $$
In particular, $pi(tilde S) = M$, and $tilde S$ will be a submanifold of $TM$ iff the dimension of $T_pM^{G_p}$ is the same for all $pin M$.



$G$ being connected won't necessarily make this happen: Another counterexample is $S^1$ acting by rotation on $S^2$, fixing the north and south poles, $n$ and $s$. $tilde S$ is then $T(S^2setminus{n,s})cup {n,s}$, and the points $n$ and $s$ do not have neighborhoods homeomorphic to open balls.



EDIT



To address the modified question: If $pin S$, then the orbit $Gp$ has a neighborhood homeomorphic to $Gtimes_{G_p} mathcal H_p$ and we're assuming that $mathcal H_p^{G_p} neq 0$. If $qin mathcal H_p$ is any point, then $mathcal H_p^{G_p}$ will be fixed by $G_q$ and still be orthogonal to $Gq$, hence $q$ and every point in $Gq$ will be in $S$. This shows that a neighborhood of $Gp$ is contained in $S$. Thus $S$ is an open submanifold of $M$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    what if we change a little bit my definition by asking the following: $S = {p in M : exists 0 neq X in mathcal {H}_p subset T_pM : G_X = G_p}$? Now I ask if such $S$ is a submanifold of $M$, where $cal H_p$ is the space $g$-orthogonal to $T_pGcdot p$.
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 23:13








  • 1




    $begingroup$
    My first reaction is that, yes, $S$ will be a submanifold. In fact, an open submanifold: If $pin S$ then all points in an open neighborhood of $p$ should also satisfy your new condition. Essentially, a neighborhood of $p$ will look like $T_p M$ with the action of $G_p$.
    $endgroup$
    – Steve Costenoble
    Mar 10 at 0:11










  • $begingroup$
    I have edited the question in order to encompass $S$ to it, it would be extremely helpful if you explain to me your thoughts on why $S$ has a chance to be a submanifold.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 0:45












  • $begingroup$
    I will accept your answer since I could prove it was true following your comment, I will post the proof on the question.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 18:45
















4












$begingroup$

Your definition implies that
$$ tilde S = bigcup_{pin M} T_pM^{G_p}. $$
In particular, $pi(tilde S) = M$, and $tilde S$ will be a submanifold of $TM$ iff the dimension of $T_pM^{G_p}$ is the same for all $pin M$.



$G$ being connected won't necessarily make this happen: Another counterexample is $S^1$ acting by rotation on $S^2$, fixing the north and south poles, $n$ and $s$. $tilde S$ is then $T(S^2setminus{n,s})cup {n,s}$, and the points $n$ and $s$ do not have neighborhoods homeomorphic to open balls.



EDIT



To address the modified question: If $pin S$, then the orbit $Gp$ has a neighborhood homeomorphic to $Gtimes_{G_p} mathcal H_p$ and we're assuming that $mathcal H_p^{G_p} neq 0$. If $qin mathcal H_p$ is any point, then $mathcal H_p^{G_p}$ will be fixed by $G_q$ and still be orthogonal to $Gq$, hence $q$ and every point in $Gq$ will be in $S$. This shows that a neighborhood of $Gp$ is contained in $S$. Thus $S$ is an open submanifold of $M$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    what if we change a little bit my definition by asking the following: $S = {p in M : exists 0 neq X in mathcal {H}_p subset T_pM : G_X = G_p}$? Now I ask if such $S$ is a submanifold of $M$, where $cal H_p$ is the space $g$-orthogonal to $T_pGcdot p$.
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 23:13








  • 1




    $begingroup$
    My first reaction is that, yes, $S$ will be a submanifold. In fact, an open submanifold: If $pin S$ then all points in an open neighborhood of $p$ should also satisfy your new condition. Essentially, a neighborhood of $p$ will look like $T_p M$ with the action of $G_p$.
    $endgroup$
    – Steve Costenoble
    Mar 10 at 0:11










  • $begingroup$
    I have edited the question in order to encompass $S$ to it, it would be extremely helpful if you explain to me your thoughts on why $S$ has a chance to be a submanifold.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 0:45












  • $begingroup$
    I will accept your answer since I could prove it was true following your comment, I will post the proof on the question.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 18:45














4












4








4





$begingroup$

Your definition implies that
$$ tilde S = bigcup_{pin M} T_pM^{G_p}. $$
In particular, $pi(tilde S) = M$, and $tilde S$ will be a submanifold of $TM$ iff the dimension of $T_pM^{G_p}$ is the same for all $pin M$.



$G$ being connected won't necessarily make this happen: Another counterexample is $S^1$ acting by rotation on $S^2$, fixing the north and south poles, $n$ and $s$. $tilde S$ is then $T(S^2setminus{n,s})cup {n,s}$, and the points $n$ and $s$ do not have neighborhoods homeomorphic to open balls.



EDIT



To address the modified question: If $pin S$, then the orbit $Gp$ has a neighborhood homeomorphic to $Gtimes_{G_p} mathcal H_p$ and we're assuming that $mathcal H_p^{G_p} neq 0$. If $qin mathcal H_p$ is any point, then $mathcal H_p^{G_p}$ will be fixed by $G_q$ and still be orthogonal to $Gq$, hence $q$ and every point in $Gq$ will be in $S$. This shows that a neighborhood of $Gp$ is contained in $S$. Thus $S$ is an open submanifold of $M$.






share|cite|improve this answer











$endgroup$



Your definition implies that
$$ tilde S = bigcup_{pin M} T_pM^{G_p}. $$
In particular, $pi(tilde S) = M$, and $tilde S$ will be a submanifold of $TM$ iff the dimension of $T_pM^{G_p}$ is the same for all $pin M$.



$G$ being connected won't necessarily make this happen: Another counterexample is $S^1$ acting by rotation on $S^2$, fixing the north and south poles, $n$ and $s$. $tilde S$ is then $T(S^2setminus{n,s})cup {n,s}$, and the points $n$ and $s$ do not have neighborhoods homeomorphic to open balls.



EDIT



To address the modified question: If $pin S$, then the orbit $Gp$ has a neighborhood homeomorphic to $Gtimes_{G_p} mathcal H_p$ and we're assuming that $mathcal H_p^{G_p} neq 0$. If $qin mathcal H_p$ is any point, then $mathcal H_p^{G_p}$ will be fixed by $G_q$ and still be orthogonal to $Gq$, hence $q$ and every point in $Gq$ will be in $S$. This shows that a neighborhood of $Gp$ is contained in $S$. Thus $S$ is an open submanifold of $M$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Mar 10 at 3:54

























answered Mar 9 at 23:01









Steve CostenobleSteve Costenoble

1,0601514




1,0601514












  • $begingroup$
    what if we change a little bit my definition by asking the following: $S = {p in M : exists 0 neq X in mathcal {H}_p subset T_pM : G_X = G_p}$? Now I ask if such $S$ is a submanifold of $M$, where $cal H_p$ is the space $g$-orthogonal to $T_pGcdot p$.
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 23:13








  • 1




    $begingroup$
    My first reaction is that, yes, $S$ will be a submanifold. In fact, an open submanifold: If $pin S$ then all points in an open neighborhood of $p$ should also satisfy your new condition. Essentially, a neighborhood of $p$ will look like $T_p M$ with the action of $G_p$.
    $endgroup$
    – Steve Costenoble
    Mar 10 at 0:11










  • $begingroup$
    I have edited the question in order to encompass $S$ to it, it would be extremely helpful if you explain to me your thoughts on why $S$ has a chance to be a submanifold.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 0:45












  • $begingroup$
    I will accept your answer since I could prove it was true following your comment, I will post the proof on the question.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 18:45


















  • $begingroup$
    what if we change a little bit my definition by asking the following: $S = {p in M : exists 0 neq X in mathcal {H}_p subset T_pM : G_X = G_p}$? Now I ask if such $S$ is a submanifold of $M$, where $cal H_p$ is the space $g$-orthogonal to $T_pGcdot p$.
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 23:13








  • 1




    $begingroup$
    My first reaction is that, yes, $S$ will be a submanifold. In fact, an open submanifold: If $pin S$ then all points in an open neighborhood of $p$ should also satisfy your new condition. Essentially, a neighborhood of $p$ will look like $T_p M$ with the action of $G_p$.
    $endgroup$
    – Steve Costenoble
    Mar 10 at 0:11










  • $begingroup$
    I have edited the question in order to encompass $S$ to it, it would be extremely helpful if you explain to me your thoughts on why $S$ has a chance to be a submanifold.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 0:45












  • $begingroup$
    I will accept your answer since I could prove it was true following your comment, I will post the proof on the question.
    $endgroup$
    – L.F. Cavenaghi
    Mar 10 at 18:45
















$begingroup$
what if we change a little bit my definition by asking the following: $S = {p in M : exists 0 neq X in mathcal {H}_p subset T_pM : G_X = G_p}$? Now I ask if such $S$ is a submanifold of $M$, where $cal H_p$ is the space $g$-orthogonal to $T_pGcdot p$.
$endgroup$
– L.F. Cavenaghi
Mar 9 at 23:13






$begingroup$
what if we change a little bit my definition by asking the following: $S = {p in M : exists 0 neq X in mathcal {H}_p subset T_pM : G_X = G_p}$? Now I ask if such $S$ is a submanifold of $M$, where $cal H_p$ is the space $g$-orthogonal to $T_pGcdot p$.
$endgroup$
– L.F. Cavenaghi
Mar 9 at 23:13






1




1




$begingroup$
My first reaction is that, yes, $S$ will be a submanifold. In fact, an open submanifold: If $pin S$ then all points in an open neighborhood of $p$ should also satisfy your new condition. Essentially, a neighborhood of $p$ will look like $T_p M$ with the action of $G_p$.
$endgroup$
– Steve Costenoble
Mar 10 at 0:11




$begingroup$
My first reaction is that, yes, $S$ will be a submanifold. In fact, an open submanifold: If $pin S$ then all points in an open neighborhood of $p$ should also satisfy your new condition. Essentially, a neighborhood of $p$ will look like $T_p M$ with the action of $G_p$.
$endgroup$
– Steve Costenoble
Mar 10 at 0:11












$begingroup$
I have edited the question in order to encompass $S$ to it, it would be extremely helpful if you explain to me your thoughts on why $S$ has a chance to be a submanifold.
$endgroup$
– L.F. Cavenaghi
Mar 10 at 0:45






$begingroup$
I have edited the question in order to encompass $S$ to it, it would be extremely helpful if you explain to me your thoughts on why $S$ has a chance to be a submanifold.
$endgroup$
– L.F. Cavenaghi
Mar 10 at 0:45














$begingroup$
I will accept your answer since I could prove it was true following your comment, I will post the proof on the question.
$endgroup$
– L.F. Cavenaghi
Mar 10 at 18:45




$begingroup$
I will accept your answer since I could prove it was true following your comment, I will post the proof on the question.
$endgroup$
– L.F. Cavenaghi
Mar 10 at 18:45











2












$begingroup$

For the obvious reflection action of $mathbb{Z}/2mathbb{Z}$ on $S^1$ this set $tilde S $ has two singular points at the zero vectors tangent at points $p=(1,0)$ and $q=(-1,0)$.
Then $tilde S$ is $T(S^1setminus {p,q}) cup {p_0,q_0}$ where $p_0, q_0$ are zero vectors at $p,q$. This is a connected set and after removing $p_0,q_0$ we obtain a disconnected set. So obviously it is not a manifold because every connected manifold of dimension at least $2$, remains connected after removing a finite set.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    "every connected manifold remains connected after removing a finite set" -- this is not true for 1-dimensional manifolds. Nonetheless, your example is valid: if $x$ is one of the fixed points of the reflection action, then a neighborhood of $(x,0)intilde S$ is homeomorphic to $mathbb R^2$ minus the coordinate axes but including the origin, so it cannot be a manifold.
    $endgroup$
    – Arun Debray
    Mar 9 at 21:42










  • $begingroup$
    @ArunDebray yes thank you but $tilde S $ is 2 dimensional at generic points. I revise the answer. Thanks again for your correction!
    $endgroup$
    – Ali Taghavi
    Mar 9 at 21:44






  • 1




    $begingroup$
    @AliTaghavi, thank you very much. Do you know if one can restrict some hypothesis in order to obtain a manifold? Or, what kind of structure this set has?
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 22:18










  • $begingroup$
    You are well come. What about if we assume $G$ is connected? I am not sure what is the answer in this case.
    $endgroup$
    – Ali Taghavi
    Mar 9 at 22:29






  • 1




    $begingroup$
    @AliTaghavi, take a look to the other answer and possible on the comments. I changed a little the problem.
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 23:15
















2












$begingroup$

For the obvious reflection action of $mathbb{Z}/2mathbb{Z}$ on $S^1$ this set $tilde S $ has two singular points at the zero vectors tangent at points $p=(1,0)$ and $q=(-1,0)$.
Then $tilde S$ is $T(S^1setminus {p,q}) cup {p_0,q_0}$ where $p_0, q_0$ are zero vectors at $p,q$. This is a connected set and after removing $p_0,q_0$ we obtain a disconnected set. So obviously it is not a manifold because every connected manifold of dimension at least $2$, remains connected after removing a finite set.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    "every connected manifold remains connected after removing a finite set" -- this is not true for 1-dimensional manifolds. Nonetheless, your example is valid: if $x$ is one of the fixed points of the reflection action, then a neighborhood of $(x,0)intilde S$ is homeomorphic to $mathbb R^2$ minus the coordinate axes but including the origin, so it cannot be a manifold.
    $endgroup$
    – Arun Debray
    Mar 9 at 21:42










  • $begingroup$
    @ArunDebray yes thank you but $tilde S $ is 2 dimensional at generic points. I revise the answer. Thanks again for your correction!
    $endgroup$
    – Ali Taghavi
    Mar 9 at 21:44






  • 1




    $begingroup$
    @AliTaghavi, thank you very much. Do you know if one can restrict some hypothesis in order to obtain a manifold? Or, what kind of structure this set has?
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 22:18










  • $begingroup$
    You are well come. What about if we assume $G$ is connected? I am not sure what is the answer in this case.
    $endgroup$
    – Ali Taghavi
    Mar 9 at 22:29






  • 1




    $begingroup$
    @AliTaghavi, take a look to the other answer and possible on the comments. I changed a little the problem.
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 23:15














2












2








2





$begingroup$

For the obvious reflection action of $mathbb{Z}/2mathbb{Z}$ on $S^1$ this set $tilde S $ has two singular points at the zero vectors tangent at points $p=(1,0)$ and $q=(-1,0)$.
Then $tilde S$ is $T(S^1setminus {p,q}) cup {p_0,q_0}$ where $p_0, q_0$ are zero vectors at $p,q$. This is a connected set and after removing $p_0,q_0$ we obtain a disconnected set. So obviously it is not a manifold because every connected manifold of dimension at least $2$, remains connected after removing a finite set.






share|cite|improve this answer











$endgroup$



For the obvious reflection action of $mathbb{Z}/2mathbb{Z}$ on $S^1$ this set $tilde S $ has two singular points at the zero vectors tangent at points $p=(1,0)$ and $q=(-1,0)$.
Then $tilde S$ is $T(S^1setminus {p,q}) cup {p_0,q_0}$ where $p_0, q_0$ are zero vectors at $p,q$. This is a connected set and after removing $p_0,q_0$ we obtain a disconnected set. So obviously it is not a manifold because every connected manifold of dimension at least $2$, remains connected after removing a finite set.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Mar 9 at 21:45

























answered Mar 9 at 20:36









Ali TaghaviAli Taghavi

10252085




10252085








  • 1




    $begingroup$
    "every connected manifold remains connected after removing a finite set" -- this is not true for 1-dimensional manifolds. Nonetheless, your example is valid: if $x$ is one of the fixed points of the reflection action, then a neighborhood of $(x,0)intilde S$ is homeomorphic to $mathbb R^2$ minus the coordinate axes but including the origin, so it cannot be a manifold.
    $endgroup$
    – Arun Debray
    Mar 9 at 21:42










  • $begingroup$
    @ArunDebray yes thank you but $tilde S $ is 2 dimensional at generic points. I revise the answer. Thanks again for your correction!
    $endgroup$
    – Ali Taghavi
    Mar 9 at 21:44






  • 1




    $begingroup$
    @AliTaghavi, thank you very much. Do you know if one can restrict some hypothesis in order to obtain a manifold? Or, what kind of structure this set has?
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 22:18










  • $begingroup$
    You are well come. What about if we assume $G$ is connected? I am not sure what is the answer in this case.
    $endgroup$
    – Ali Taghavi
    Mar 9 at 22:29






  • 1




    $begingroup$
    @AliTaghavi, take a look to the other answer and possible on the comments. I changed a little the problem.
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 23:15














  • 1




    $begingroup$
    "every connected manifold remains connected after removing a finite set" -- this is not true for 1-dimensional manifolds. Nonetheless, your example is valid: if $x$ is one of the fixed points of the reflection action, then a neighborhood of $(x,0)intilde S$ is homeomorphic to $mathbb R^2$ minus the coordinate axes but including the origin, so it cannot be a manifold.
    $endgroup$
    – Arun Debray
    Mar 9 at 21:42










  • $begingroup$
    @ArunDebray yes thank you but $tilde S $ is 2 dimensional at generic points. I revise the answer. Thanks again for your correction!
    $endgroup$
    – Ali Taghavi
    Mar 9 at 21:44






  • 1




    $begingroup$
    @AliTaghavi, thank you very much. Do you know if one can restrict some hypothesis in order to obtain a manifold? Or, what kind of structure this set has?
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 22:18










  • $begingroup$
    You are well come. What about if we assume $G$ is connected? I am not sure what is the answer in this case.
    $endgroup$
    – Ali Taghavi
    Mar 9 at 22:29






  • 1




    $begingroup$
    @AliTaghavi, take a look to the other answer and possible on the comments. I changed a little the problem.
    $endgroup$
    – L.F. Cavenaghi
    Mar 9 at 23:15








1




1




$begingroup$
"every connected manifold remains connected after removing a finite set" -- this is not true for 1-dimensional manifolds. Nonetheless, your example is valid: if $x$ is one of the fixed points of the reflection action, then a neighborhood of $(x,0)intilde S$ is homeomorphic to $mathbb R^2$ minus the coordinate axes but including the origin, so it cannot be a manifold.
$endgroup$
– Arun Debray
Mar 9 at 21:42




$begingroup$
"every connected manifold remains connected after removing a finite set" -- this is not true for 1-dimensional manifolds. Nonetheless, your example is valid: if $x$ is one of the fixed points of the reflection action, then a neighborhood of $(x,0)intilde S$ is homeomorphic to $mathbb R^2$ minus the coordinate axes but including the origin, so it cannot be a manifold.
$endgroup$
– Arun Debray
Mar 9 at 21:42












$begingroup$
@ArunDebray yes thank you but $tilde S $ is 2 dimensional at generic points. I revise the answer. Thanks again for your correction!
$endgroup$
– Ali Taghavi
Mar 9 at 21:44




$begingroup$
@ArunDebray yes thank you but $tilde S $ is 2 dimensional at generic points. I revise the answer. Thanks again for your correction!
$endgroup$
– Ali Taghavi
Mar 9 at 21:44




1




1




$begingroup$
@AliTaghavi, thank you very much. Do you know if one can restrict some hypothesis in order to obtain a manifold? Or, what kind of structure this set has?
$endgroup$
– L.F. Cavenaghi
Mar 9 at 22:18




$begingroup$
@AliTaghavi, thank you very much. Do you know if one can restrict some hypothesis in order to obtain a manifold? Or, what kind of structure this set has?
$endgroup$
– L.F. Cavenaghi
Mar 9 at 22:18












$begingroup$
You are well come. What about if we assume $G$ is connected? I am not sure what is the answer in this case.
$endgroup$
– Ali Taghavi
Mar 9 at 22:29




$begingroup$
You are well come. What about if we assume $G$ is connected? I am not sure what is the answer in this case.
$endgroup$
– Ali Taghavi
Mar 9 at 22:29




1




1




$begingroup$
@AliTaghavi, take a look to the other answer and possible on the comments. I changed a little the problem.
$endgroup$
– L.F. Cavenaghi
Mar 9 at 23:15




$begingroup$
@AliTaghavi, take a look to the other answer and possible on the comments. I changed a little the problem.
$endgroup$
– L.F. Cavenaghi
Mar 9 at 23:15


















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