The hypothesis $Kleq 0$ in the proof of Hadamard's Theorem
In chapter 7 from do Carmo's Riemannian Geometry, right after proving Hadamard's theorem, there is the following remark:
When he says "poles can exist in non-compact manifolds which have positive sectional curvature", I believe he is trying to justify the hypothesis "$Kleq 0$".
What I don't understand is: why did he have to talk about "non-compact manifolds"?
I thought he would say "poles can exist in complete manifolds with positive curvature", which makes perfect sense and justifies the hypothesis "$Kleq 0$".
I know I'm not crazy because exercise 13 (where $M$ is the parabolloid $z=x^2+y^2$) is precisely an example of a complete manifold with positive curvature having a pole (namely, the origin).
So what is this "non-compact" business?
riemannian-geometry smooth-manifolds curvature
asked 17 hours ago
rmdmc89
2,0711921
add a comment |
In chapter 7 from do Carmo's Riemannian Geometry, right after proving Hadamard's theorem, there is the following remark:
When he says "poles can exist in non-compact manifolds which have positive sectional curvature", I believe he is trying to justify the hypothesis "$Kleq 0$".
What I don't understand is: why did he have to talk about "non-compact manifolds"?
I thought he would say "poles can exist in complete manifolds with positive curvature", which makes perfect sense and justifies the hypothesis "$Kleq 0$".
I know I'm not crazy because exercise 13 (where $M$ is the parabolloid $z=x^2+y^2$) is precisely an example of a complete manifold with positive curvature having a pole (namely, the origin).
So what is this "non-compact" business?
riemannian-geometry smooth-manifolds curvature
asked 17 hours ago
rmdmc89
2,0711921
If $K>0$ and $M$ is compact then there are no poles. Of course, you can still have $K=0$, $M$ compact and no poles.
– Moishe Cohen
6 hours ago
add a comment |
In chapter 7 from do Carmo's Riemannian Geometry, right after proving Hadamard's theorem, there is the following remark:
When he says "poles can exist in non-compact manifolds which have positive sectional curvature", I believe he is trying to justify the hypothesis "$Kleq 0$".
What I don't understand is: why did he have to talk about "non-compact manifolds"?
I thought he would say "poles can exist in complete manifolds with positive curvature", which makes perfect sense and justifies the hypothesis "$Kleq 0$".
I know I'm not crazy because exercise 13 (where $M$ is the parabolloid $z=x^2+y^2$) is precisely an example of a complete manifold with positive curvature having a pole (namely, the origin).
So what is this "non-compact" business?
riemannian-geometry smooth-manifolds curvature
asked 17 hours ago
rmdmc89
2,0711921
In chapter 7 from do Carmo's Riemannian Geometry, right after proving Hadamard's theorem, there is the following remark:
When he says "poles can exist in non-compact manifolds which have positive sectional curvature", I believe he is trying to justify the hypothesis "$Kleq 0$".
What I don't understand is: why did he have to talk about "non-compact manifolds"?
I thought he would say "poles can exist in complete manifolds with positive curvature", which makes perfect sense and justifies the hypothesis "$Kleq 0$".
I know I'm not crazy because exercise 13 (where $M$ is the parabolloid $z=x^2+y^2$) is precisely an example of a complete manifold with positive curvature having a pole (namely, the origin).
So what is this "non-compact" business?
riemannian-geometry smooth-manifolds curvature
riemannian-geometry smooth-manifolds curvature
asked 17 hours ago
rmdmc89
2,0711921
asked 17 hours ago
rmdmc89
2,0711921
edited 17 hours ago
asked 17 hours ago
rmdmc89
2,0711921
asked 17 hours ago
rmdmc89
2,0711921
asked 17 hours ago
rmdmc89
2,0711921
2,0711921
If $K>0$ and $M$ is compact then there are no poles. Of course, you can still have $K=0$, $M$ compact and no poles.
– Moishe Cohen
6 hours ago
add a comment |
If $K>0$ and $M$ is compact then there are no poles. Of course, you can still have $K=0$, $M$ compact and no poles.
– Moishe Cohen
6 hours ago
If $K>0$ and $M$ is compact then there are no poles. Of course, you can still have $K=0$, $M$ compact and no poles.
– Moishe Cohen
6 hours ago
If $K>0$ and $M$ is compact then there are no poles. Of course, you can still have $K=0$, $M$ compact and no poles.
– Moishe Cohen
6 hours ago
add a comment |
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If $K>0$ and $M$ is compact then there are no poles. Of course, you can still have $K=0$, $M$ compact and no poles.
– Moishe Cohen
6 hours ago