Mixing
Poincaré–Hopf and Morse inequalities
$begingroup$
Disclaimer: I am not a differential geometer, so maybe this question does not make sense:
Let $(M^m,g)$ be a Riemannian manifold and $f:Mto mathbb{R}$ a Morse function. Since $g$ is pointwise non-degenerate, a obtain a vector field by $X_p:=g_p^{-1}(mathrm{d}f|_p)$. I have two notions of “critical point” and “index”:
The Morse theoric one: $pin M$ is critical if $mathrm{d}f|_p=0$. The index $I(p)$ is the negative sign of $mathrm{d}^2f|_p$
The one for Poincaré–Hopf: $pin M$ is critical if $X_p=0$. The index $mathrm{ind}_p(X)$ is the degree of the following map, locally around $p$ (after choosing charts and obtaining a local trivialisation of $TM$):
$$varphi:mathbb{S}^{m-1}to mathbb{S}^{m-1}, qmapsto frac{X(q)}{|X(q)|}.$$
Obviously, the two notions of being critical are equivalent. What about the indices? They are not the same as $0le I(p)le m$, but not necessarily $0le mathrm{ind}_p(X)le m$. In what way are they related?
The only idea I had was the combination of Poincaré–Hopf and Morse theory (correct?):
$$sum_{text{critical $p$}} mathrm{ind}_p(X) = chi(M) = sum_{text{critical $p$}} (-1)^{I(p)}.$$
differential-topology riemannian-geometry morse-theory
asked Jan 16 at 11:09
FKranholdFKranhold
1987
$endgroup$
add a comment |
$begingroup$
Disclaimer: I am not a differential geometer, so maybe this question does not make sense:
Let $(M^m,g)$ be a Riemannian manifold and $f:Mto mathbb{R}$ a Morse function. Since $g$ is pointwise non-degenerate, a obtain a vector field by $X_p:=g_p^{-1}(mathrm{d}f|_p)$. I have two notions of “critical point” and “index”:
The Morse theoric one: $pin M$ is critical if $mathrm{d}f|_p=0$. The index $I(p)$ is the negative sign of $mathrm{d}^2f|_p$
The one for Poincaré–Hopf: $pin M$ is critical if $X_p=0$. The index $mathrm{ind}_p(X)$ is the degree of the following map, locally around $p$ (after choosing charts and obtaining a local trivialisation of $TM$):
$$varphi:mathbb{S}^{m-1}to mathbb{S}^{m-1}, qmapsto frac{X(q)}{|X(q)|}.$$
Obviously, the two notions of being critical are equivalent. What about the indices? They are not the same as $0le I(p)le m$, but not necessarily $0le mathrm{ind}_p(X)le m$. In what way are they related?
The only idea I had was the combination of Poincaré–Hopf and Morse theory (correct?):
$$sum_{text{critical $p$}} mathrm{ind}_p(X) = chi(M) = sum_{text{critical $p$}} (-1)^{I(p)}.$$
differential-topology riemannian-geometry morse-theory
asked Jan 16 at 11:09
FKranholdFKranhold
1987
$endgroup$
$begingroup$
First, the adjunction operator $g_p: T_pM rightarrow T^*_pM$ is an isomorphism, so the definition of a critical point is the same.
$endgroup$
– Mindlack
Jan 16 at 11:24
3
$begingroup$
You are working too hard. This is a local calculation, and one that is essentially independent of the metric - a homotopy of metrics induces a homotopy of vector fields, all of which are transverse to 0 (aka, 'the gradient flow has nondegenerate zeroes' is a statement independent of metric). So work with the Euclidean metric on a chart, and similarly that $f$ is one of the standard quadratic Morse functions $sum pm x_i^2$. Then a quick check shows that $(-1)^{I(p)} = text{ind}_p$.
$endgroup$
– Mike Miller
Jan 16 at 14:57
add a comment |
$begingroup$
Disclaimer: I am not a differential geometer, so maybe this question does not make sense:
Let $(M^m,g)$ be a Riemannian manifold and $f:Mto mathbb{R}$ a Morse function. Since $g$ is pointwise non-degenerate, a obtain a vector field by $X_p:=g_p^{-1}(mathrm{d}f|_p)$. I have two notions of “critical point” and “index”:
The Morse theoric one: $pin M$ is critical if $mathrm{d}f|_p=0$. The index $I(p)$ is the negative sign of $mathrm{d}^2f|_p$
The one for Poincaré–Hopf: $pin M$ is critical if $X_p=0$. The index $mathrm{ind}_p(X)$ is the degree of the following map, locally around $p$ (after choosing charts and obtaining a local trivialisation of $TM$):
$$varphi:mathbb{S}^{m-1}to mathbb{S}^{m-1}, qmapsto frac{X(q)}{|X(q)|}.$$
Obviously, the two notions of being critical are equivalent. What about the indices? They are not the same as $0le I(p)le m$, but not necessarily $0le mathrm{ind}_p(X)le m$. In what way are they related?
The only idea I had was the combination of Poincaré–Hopf and Morse theory (correct?):
$$sum_{text{critical $p$}} mathrm{ind}_p(X) = chi(M) = sum_{text{critical $p$}} (-1)^{I(p)}.$$
differential-topology riemannian-geometry morse-theory
asked Jan 16 at 11:09
FKranholdFKranhold
1987
$endgroup$
Disclaimer: I am not a differential geometer, so maybe this question does not make sense:
Let $(M^m,g)$ be a Riemannian manifold and $f:Mto mathbb{R}$ a Morse function. Since $g$ is pointwise non-degenerate, a obtain a vector field by $X_p:=g_p^{-1}(mathrm{d}f|_p)$. I have two notions of “critical point” and “index”:
The Morse theoric one: $pin M$ is critical if $mathrm{d}f|_p=0$. The index $I(p)$ is the negative sign of $mathrm{d}^2f|_p$
The one for Poincaré–Hopf: $pin M$ is critical if $X_p=0$. The index $mathrm{ind}_p(X)$ is the degree of the following map, locally around $p$ (after choosing charts and obtaining a local trivialisation of $TM$):
$$varphi:mathbb{S}^{m-1}to mathbb{S}^{m-1}, qmapsto frac{X(q)}{|X(q)|}.$$
Obviously, the two notions of being critical are equivalent. What about the indices? They are not the same as $0le I(p)le m$, but not necessarily $0le mathrm{ind}_p(X)le m$. In what way are they related?
The only idea I had was the combination of Poincaré–Hopf and Morse theory (correct?):
$$sum_{text{critical $p$}} mathrm{ind}_p(X) = chi(M) = sum_{text{critical $p$}} (-1)^{I(p)}.$$
differential-topology riemannian-geometry morse-theory
differential-topology riemannian-geometry morse-theory
asked Jan 16 at 11:09
FKranholdFKranhold
1987
asked Jan 16 at 11:09
FKranholdFKranhold
1987
asked Jan 16 at 11:09
FKranholdFKranhold
1987
asked Jan 16 at 11:09
FKranholdFKranhold
1987
asked Jan 16 at 11:09
FKranholdFKranhold
1987
1987
$begingroup$
First, the adjunction operator $g_p: T_pM rightarrow T^*_pM$ is an isomorphism, so the definition of a critical point is the same.
$endgroup$
– Mindlack
Jan 16 at 11:24
3
$begingroup$
You are working too hard. This is a local calculation, and one that is essentially independent of the metric - a homotopy of metrics induces a homotopy of vector fields, all of which are transverse to 0 (aka, 'the gradient flow has nondegenerate zeroes' is a statement independent of metric). So work with the Euclidean metric on a chart, and similarly that $f$ is one of the standard quadratic Morse functions $sum pm x_i^2$. Then a quick check shows that $(-1)^{I(p)} = text{ind}_p$.
$endgroup$
– Mike Miller
Jan 16 at 14:57
add a comment |
$begingroup$
First, the adjunction operator $g_p: T_pM rightarrow T^*_pM$ is an isomorphism, so the definition of a critical point is the same.
$endgroup$
– Mindlack
Jan 16 at 11:24
3
$begingroup$
You are working too hard. This is a local calculation, and one that is essentially independent of the metric - a homotopy of metrics induces a homotopy of vector fields, all of which are transverse to 0 (aka, 'the gradient flow has nondegenerate zeroes' is a statement independent of metric). So work with the Euclidean metric on a chart, and similarly that $f$ is one of the standard quadratic Morse functions $sum pm x_i^2$. Then a quick check shows that $(-1)^{I(p)} = text{ind}_p$.
$endgroup$
– Mike Miller
Jan 16 at 14:57
$begingroup$
First, the adjunction operator $g_p: T_pM rightarrow T^*_pM$ is an isomorphism, so the definition of a critical point is the same.
$endgroup$
– Mindlack
Jan 16 at 11:24
$begingroup$
First, the adjunction operator $g_p: T_pM rightarrow T^*_pM$ is an isomorphism, so the definition of a critical point is the same.
$endgroup$
– Mindlack
Jan 16 at 11:24
3
3
$begingroup$
You are working too hard. This is a local calculation, and one that is essentially independent of the metric - a homotopy of metrics induces a homotopy of vector fields, all of which are transverse to 0 (aka, 'the gradient flow has nondegenerate zeroes' is a statement independent of metric). So work with the Euclidean metric on a chart, and similarly that $f$ is one of the standard quadratic Morse functions $sum pm x_i^2$. Then a quick check shows that $(-1)^{I(p)} = text{ind}_p$.
$endgroup$
– Mike Miller
Jan 16 at 14:57
$begingroup$
You are working too hard. This is a local calculation, and one that is essentially independent of the metric - a homotopy of metrics induces a homotopy of vector fields, all of which are transverse to 0 (aka, 'the gradient flow has nondegenerate zeroes' is a statement independent of metric). So work with the Euclidean metric on a chart, and similarly that $f$ is one of the standard quadratic Morse functions $sum pm x_i^2$. Then a quick check shows that $(-1)^{I(p)} = text{ind}_p$.
$endgroup$
– Mike Miller
Jan 16 at 14:57
add a comment |
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$begingroup$
First, the adjunction operator $g_p: T_pM rightarrow T^*_pM$ is an isomorphism, so the definition of a critical point is the same.
$endgroup$
– Mindlack
Jan 16 at 11:24
3
$begingroup$
You are working too hard. This is a local calculation, and one that is essentially independent of the metric - a homotopy of metrics induces a homotopy of vector fields, all of which are transverse to 0 (aka, 'the gradient flow has nondegenerate zeroes' is a statement independent of metric). So work with the Euclidean metric on a chart, and similarly that $f$ is one of the standard quadratic Morse functions $sum pm x_i^2$. Then a quick check shows that $(-1)^{I(p)} = text{ind}_p$.
$endgroup$
– Mike Miller
Jan 16 at 14:57