Hadamard theorem about embedding
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The following theorem is commonly attributed to Hadamard.
Assume $Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $Sigma$ is embedded and bounds a convex set.
Many authors refer to Hadamard's Sur certaines propriétés des trajectoires en Dynamique (1897)
(for example, J.J.Stoker in his Über die Gestalt der positiv... (1936)).
Likely the statement is there, but the paper is long, it is in French and often the statements are not clearly marked; I was searching for it for several days. I asked a friend and she said that it was there 20 years ago, but she could not find it; she also said that it was not easy to extract it from what is written ( = one has to think). [For sure the word immersion is not there.]
I hope someone here knows this paper and can help me.
reference-request dg.differential-geometry curves-and-surfaces surfaces
asked yesterday
Anton Petrunin
26.2k578196
add a comment |
up vote
12
down vote
favorite
The following theorem is commonly attributed to Hadamard.
Assume $Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $Sigma$ is embedded and bounds a convex set.
Many authors refer to Hadamard's Sur certaines propriétés des trajectoires en Dynamique (1897)
(for example, J.J.Stoker in his Über die Gestalt der positiv... (1936)).
Likely the statement is there, but the paper is long, it is in French and often the statements are not clearly marked; I was searching for it for several days. I asked a friend and she said that it was there 20 years ago, but she could not find it; she also said that it was not easy to extract it from what is written ( = one has to think). [For sure the word immersion is not there.]
I hope someone here knows this paper and can help me.
reference-request dg.differential-geometry curves-and-surfaces surfaces
asked yesterday
Anton Petrunin
26.2k578196
1
To remain in the spirit of this site, a present-day referee would probably tell Hadamard : "unclear what you're claiming" !
– Sylvain JULIEN
yesterday
add a comment |
up vote
12
down vote
favorite
up vote
12
down vote
favorite
The following theorem is commonly attributed to Hadamard.
Assume $Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $Sigma$ is embedded and bounds a convex set.
Many authors refer to Hadamard's Sur certaines propriétés des trajectoires en Dynamique (1897)
(for example, J.J.Stoker in his Über die Gestalt der positiv... (1936)).
Likely the statement is there, but the paper is long, it is in French and often the statements are not clearly marked; I was searching for it for several days. I asked a friend and she said that it was there 20 years ago, but she could not find it; she also said that it was not easy to extract it from what is written ( = one has to think). [For sure the word immersion is not there.]
I hope someone here knows this paper and can help me.
reference-request dg.differential-geometry curves-and-surfaces surfaces
asked yesterday
Anton Petrunin
26.2k578196
The following theorem is commonly attributed to Hadamard.
Assume $Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $Sigma$ is embedded and bounds a convex set.
Many authors refer to Hadamard's Sur certaines propriétés des trajectoires en Dynamique (1897)
(for example, J.J.Stoker in his Über die Gestalt der positiv... (1936)).
Likely the statement is there, but the paper is long, it is in French and often the statements are not clearly marked; I was searching for it for several days. I asked a friend and she said that it was there 20 years ago, but she could not find it; she also said that it was not easy to extract it from what is written ( = one has to think). [For sure the word immersion is not there.]
I hope someone here knows this paper and can help me.
reference-request dg.differential-geometry curves-and-surfaces surfaces
reference-request dg.differential-geometry curves-and-surfaces surfaces
asked yesterday
Anton Petrunin
26.2k578196
asked yesterday
Anton Petrunin
26.2k578196
edited 23 hours ago
asked yesterday
Anton Petrunin
26.2k578196
asked yesterday
Anton Petrunin
26.2k578196
asked yesterday
Anton Petrunin
26.2k578196
26.2k578196
1
To remain in the spirit of this site, a present-day referee would probably tell Hadamard : "unclear what you're claiming" !
– Sylvain JULIEN
yesterday
add a comment |
1
To remain in the spirit of this site, a present-day referee would probably tell Hadamard : "unclear what you're claiming" !
– Sylvain JULIEN
yesterday
1
1
To remain in the spirit of this site, a present-day referee would probably tell Hadamard : "unclear what you're claiming" !
– Sylvain JULIEN
yesterday
To remain in the spirit of this site, a present-day referee would probably tell Hadamard : "unclear what you're claiming" !
– Sylvain JULIEN
yesterday
add a comment |
1 Answer
1
active
oldest
votes
up vote
8
down vote
accepted
I think the relevant location is item 23, page 352, but what Hadamard aims to is stated as follows:
A smooth, co-orientable surface of $mathbb{R}^3$ with Gauss curvature bounded below by some $kappa >0$ is simply connected. (implicitly, the surface is compact without boundary)
("Or une surface à deux côtés et sans points singuliers, à courbure partout positive (la valeur zéro et les valeurs infiniment petites étant exclues) est toujours simplement connexe.")
The goal is to use the Gauss-Bonnet Formula to deduce that when curvature is positive, any two closed geodesics must meet (otherwise they would together bound a total curvature 0 region of the surface).
What is not clear from the text of item 23 is whether the surface assumed to be immersed or embedded. He basically says that the normal map is a global diffeomorphism, because positive curvature makes it a covering of the sphere. It seems the argument does provide the statement attributed to this paper, although it seems not explicitly stated.
answered yesterday
Benoît Kloeckner
10.8k14382
The arguments in 23 do not seem to show that immersed sphere is embedded, even informally; am I wrong? [I see also pictures on page 379 which are relevant to a proof I know, but the words around these pictures seem to be irrelevant.]
– Anton Petrunin
yesterday
@AntonPetrunin you are probably right but I do not have much time checking in details. I would not be surprised, given the informality of the discussion, if the attribution of this statement would be somewhat of a stretch. In any case, I do not think it was the point Hadamard wanted to make (and he might assume the embedding in the first place).
– Benoît Kloeckner
15 hours ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
8
down vote
accepted
I think the relevant location is item 23, page 352, but what Hadamard aims to is stated as follows:
A smooth, co-orientable surface of $mathbb{R}^3$ with Gauss curvature bounded below by some $kappa >0$ is simply connected. (implicitly, the surface is compact without boundary)
("Or une surface à deux côtés et sans points singuliers, à courbure partout positive (la valeur zéro et les valeurs infiniment petites étant exclues) est toujours simplement connexe.")
The goal is to use the Gauss-Bonnet Formula to deduce that when curvature is positive, any two closed geodesics must meet (otherwise they would together bound a total curvature 0 region of the surface).
What is not clear from the text of item 23 is whether the surface assumed to be immersed or embedded. He basically says that the normal map is a global diffeomorphism, because positive curvature makes it a covering of the sphere. It seems the argument does provide the statement attributed to this paper, although it seems not explicitly stated.
answered yesterday
Benoît Kloeckner
10.8k14382
The arguments in 23 do not seem to show that immersed sphere is embedded, even informally; am I wrong? [I see also pictures on page 379 which are relevant to a proof I know, but the words around these pictures seem to be irrelevant.]
– Anton Petrunin
yesterday
@AntonPetrunin you are probably right but I do not have much time checking in details. I would not be surprised, given the informality of the discussion, if the attribution of this statement would be somewhat of a stretch. In any case, I do not think it was the point Hadamard wanted to make (and he might assume the embedding in the first place).
– Benoît Kloeckner
15 hours ago
add a comment |
up vote
8
down vote
accepted
I think the relevant location is item 23, page 352, but what Hadamard aims to is stated as follows:
A smooth, co-orientable surface of $mathbb{R}^3$ with Gauss curvature bounded below by some $kappa >0$ is simply connected. (implicitly, the surface is compact without boundary)
("Or une surface à deux côtés et sans points singuliers, à courbure partout positive (la valeur zéro et les valeurs infiniment petites étant exclues) est toujours simplement connexe.")
The goal is to use the Gauss-Bonnet Formula to deduce that when curvature is positive, any two closed geodesics must meet (otherwise they would together bound a total curvature 0 region of the surface).
What is not clear from the text of item 23 is whether the surface assumed to be immersed or embedded. He basically says that the normal map is a global diffeomorphism, because positive curvature makes it a covering of the sphere. It seems the argument does provide the statement attributed to this paper, although it seems not explicitly stated.
answered yesterday
Benoît Kloeckner
10.8k14382
The arguments in 23 do not seem to show that immersed sphere is embedded, even informally; am I wrong? [I see also pictures on page 379 which are relevant to a proof I know, but the words around these pictures seem to be irrelevant.]
– Anton Petrunin
yesterday
@AntonPetrunin you are probably right but I do not have much time checking in details. I would not be surprised, given the informality of the discussion, if the attribution of this statement would be somewhat of a stretch. In any case, I do not think it was the point Hadamard wanted to make (and he might assume the embedding in the first place).
– Benoît Kloeckner
15 hours ago
add a comment |
up vote
8
down vote
accepted
up vote
8
down vote
accepted
I think the relevant location is item 23, page 352, but what Hadamard aims to is stated as follows:
A smooth, co-orientable surface of $mathbb{R}^3$ with Gauss curvature bounded below by some $kappa >0$ is simply connected. (implicitly, the surface is compact without boundary)
("Or une surface à deux côtés et sans points singuliers, à courbure partout positive (la valeur zéro et les valeurs infiniment petites étant exclues) est toujours simplement connexe.")
The goal is to use the Gauss-Bonnet Formula to deduce that when curvature is positive, any two closed geodesics must meet (otherwise they would together bound a total curvature 0 region of the surface).
What is not clear from the text of item 23 is whether the surface assumed to be immersed or embedded. He basically says that the normal map is a global diffeomorphism, because positive curvature makes it a covering of the sphere. It seems the argument does provide the statement attributed to this paper, although it seems not explicitly stated.
answered yesterday
Benoît Kloeckner
10.8k14382
I think the relevant location is item 23, page 352, but what Hadamard aims to is stated as follows:
A smooth, co-orientable surface of $mathbb{R}^3$ with Gauss curvature bounded below by some $kappa >0$ is simply connected. (implicitly, the surface is compact without boundary)
("Or une surface à deux côtés et sans points singuliers, à courbure partout positive (la valeur zéro et les valeurs infiniment petites étant exclues) est toujours simplement connexe.")
The goal is to use the Gauss-Bonnet Formula to deduce that when curvature is positive, any two closed geodesics must meet (otherwise they would together bound a total curvature 0 region of the surface).
What is not clear from the text of item 23 is whether the surface assumed to be immersed or embedded. He basically says that the normal map is a global diffeomorphism, because positive curvature makes it a covering of the sphere. It seems the argument does provide the statement attributed to this paper, although it seems not explicitly stated.
answered yesterday
Benoît Kloeckner
10.8k14382
answered yesterday
Benoît Kloeckner
10.8k14382
answered yesterday
Benoît Kloeckner
10.8k14382
answered yesterday
Benoît Kloeckner
10.8k14382
10.8k14382
The arguments in 23 do not seem to show that immersed sphere is embedded, even informally; am I wrong? [I see also pictures on page 379 which are relevant to a proof I know, but the words around these pictures seem to be irrelevant.]
– Anton Petrunin
yesterday
@AntonPetrunin you are probably right but I do not have much time checking in details. I would not be surprised, given the informality of the discussion, if the attribution of this statement would be somewhat of a stretch. In any case, I do not think it was the point Hadamard wanted to make (and he might assume the embedding in the first place).
– Benoît Kloeckner
15 hours ago
add a comment |
The arguments in 23 do not seem to show that immersed sphere is embedded, even informally; am I wrong? [I see also pictures on page 379 which are relevant to a proof I know, but the words around these pictures seem to be irrelevant.]
– Anton Petrunin
yesterday
@AntonPetrunin you are probably right but I do not have much time checking in details. I would not be surprised, given the informality of the discussion, if the attribution of this statement would be somewhat of a stretch. In any case, I do not think it was the point Hadamard wanted to make (and he might assume the embedding in the first place).
– Benoît Kloeckner
15 hours ago
The arguments in 23 do not seem to show that immersed sphere is embedded, even informally; am I wrong? [I see also pictures on page 379 which are relevant to a proof I know, but the words around these pictures seem to be irrelevant.]
– Anton Petrunin
yesterday
The arguments in 23 do not seem to show that immersed sphere is embedded, even informally; am I wrong? [I see also pictures on page 379 which are relevant to a proof I know, but the words around these pictures seem to be irrelevant.]
– Anton Petrunin
yesterday
@AntonPetrunin you are probably right but I do not have much time checking in details. I would not be surprised, given the informality of the discussion, if the attribution of this statement would be somewhat of a stretch. In any case, I do not think it was the point Hadamard wanted to make (and he might assume the embedding in the first place).
– Benoît Kloeckner
15 hours ago
@AntonPetrunin you are probably right but I do not have much time checking in details. I would not be surprised, given the informality of the discussion, if the attribution of this statement would be somewhat of a stretch. In any case, I do not think it was the point Hadamard wanted to make (and he might assume the embedding in the first place).
– Benoît Kloeckner
15 hours ago
add a comment |
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To remain in the spirit of this site, a present-day referee would probably tell Hadamard : "unclear what you're claiming" !
– Sylvain JULIEN
yesterday