Is there a Möbius torus?
$begingroup$
Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted torus be indistinguishable from the normal torus in any relevant respect?
[This equivalent to the well-known Möbius strip should be called Möbius cylinder but it would have so much in common with a torus that I preferred to call it a Möbius torus.]
Embedded in Euclidean space the twisted and untwisted torus "look" the same - opposed to Möbius strip and cylinder -, the difference would be only in their intrinsic properties. But can there be such differences? And how do I specify them?
PS: I posted a follow-up question here.
geometry manifolds klein-bottle mobius-band
edited Apr 13 '17 at 12:19
Community♦
1
asked Oct 20 '12 at 17:38
Hans StrickerHans Stricker
6,23843988
$endgroup$
add a comment |
$begingroup$
Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted torus be indistinguishable from the normal torus in any relevant respect?
[This equivalent to the well-known Möbius strip should be called Möbius cylinder but it would have so much in common with a torus that I preferred to call it a Möbius torus.]
Embedded in Euclidean space the twisted and untwisted torus "look" the same - opposed to Möbius strip and cylinder -, the difference would be only in their intrinsic properties. But can there be such differences? And how do I specify them?
PS: I posted a follow-up question here.
geometry manifolds klein-bottle mobius-band
edited Apr 13 '17 at 12:19
Community♦
1
asked Oct 20 '12 at 17:38
Hans StrickerHans Stricker
6,23843988
$endgroup$
9
$begingroup$
This sounds suspiciously like the Klein bottle.
$endgroup$
– Fredrik Meyer
Oct 20 '12 at 17:49
$begingroup$
Are you looking for the Klein Bottle, which joins the ends of a cylinder in the opposite orientation to a torus?
$endgroup$
– Mark Bennet
Oct 20 '12 at 17:51
4
$begingroup$
A mere twist before identifying the ends of the "hose" again produces an ordinary torus with a locally euclidean metric, but its global conformal type has changed. If you identify the ends reversing the orientation you get a Klein bottle, which is a nonorientable surface.
$endgroup$
– Christian Blatter
Oct 20 '12 at 18:21
1
$begingroup$
Just to add to Christian's comment, a twist will be isotopic to the identity map, and so the resulting space will be homeomorphic to a regular torus. Think about it: if you can do it in R^3, it's not a Klein bottle. You need a degree -1 map on the circle boundaries to get the Klein bottle.
$endgroup$
– user641
Oct 21 '12 at 4:00
$begingroup$
@Christian Blatter: Can you please help me to understand better: the simply twisted torus (which I tried to ask for) has a locally euclidean metric (just like the Klein bottle), is homeomorphic to the regular torus, but has another conformal type. What exactly is the conformal type? Does it make sense to think of (closed) geodesics in this context?
$endgroup$
– Hans Stricker
Oct 23 '12 at 14:14
add a comment |
$begingroup$
Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted torus be indistinguishable from the normal torus in any relevant respect?
[This equivalent to the well-known Möbius strip should be called Möbius cylinder but it would have so much in common with a torus that I preferred to call it a Möbius torus.]
Embedded in Euclidean space the twisted and untwisted torus "look" the same - opposed to Möbius strip and cylinder -, the difference would be only in their intrinsic properties. But can there be such differences? And how do I specify them?
PS: I posted a follow-up question here.
geometry manifolds klein-bottle mobius-band
edited Apr 13 '17 at 12:19
Community♦
1
asked Oct 20 '12 at 17:38
Hans StrickerHans Stricker
6,23843988
$endgroup$
Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted torus be indistinguishable from the normal torus in any relevant respect?
[This equivalent to the well-known Möbius strip should be called Möbius cylinder but it would have so much in common with a torus that I preferred to call it a Möbius torus.]
Embedded in Euclidean space the twisted and untwisted torus "look" the same - opposed to Möbius strip and cylinder -, the difference would be only in their intrinsic properties. But can there be such differences? And how do I specify them?
PS: I posted a follow-up question here.
geometry manifolds klein-bottle mobius-band
geometry manifolds klein-bottle mobius-band
edited Apr 13 '17 at 12:19
Community♦
1
asked Oct 20 '12 at 17:38
Hans StrickerHans Stricker
6,23843988
edited Apr 13 '17 at 12:19
Community♦
1
asked Oct 20 '12 at 17:38
Hans StrickerHans Stricker
6,23843988
edited Apr 13 '17 at 12:19
Community♦
1
edited Apr 13 '17 at 12:19
Community♦
1
edited Apr 13 '17 at 12:19
Community♦
1
1
asked Oct 20 '12 at 17:38
Hans StrickerHans Stricker
6,23843988
asked Oct 20 '12 at 17:38
Hans StrickerHans Stricker
6,23843988
asked Oct 20 '12 at 17:38
Hans StrickerHans Stricker
6,23843988
6,23843988
9
$begingroup$
This sounds suspiciously like the Klein bottle.
$endgroup$
– Fredrik Meyer
Oct 20 '12 at 17:49
$begingroup$
Are you looking for the Klein Bottle, which joins the ends of a cylinder in the opposite orientation to a torus?
$endgroup$
– Mark Bennet
Oct 20 '12 at 17:51
4
$begingroup$
A mere twist before identifying the ends of the "hose" again produces an ordinary torus with a locally euclidean metric, but its global conformal type has changed. If you identify the ends reversing the orientation you get a Klein bottle, which is a nonorientable surface.
$endgroup$
– Christian Blatter
Oct 20 '12 at 18:21
1
$begingroup$
Just to add to Christian's comment, a twist will be isotopic to the identity map, and so the resulting space will be homeomorphic to a regular torus. Think about it: if you can do it in R^3, it's not a Klein bottle. You need a degree -1 map on the circle boundaries to get the Klein bottle.
$endgroup$
– user641
Oct 21 '12 at 4:00
$begingroup$
@Christian Blatter: Can you please help me to understand better: the simply twisted torus (which I tried to ask for) has a locally euclidean metric (just like the Klein bottle), is homeomorphic to the regular torus, but has another conformal type. What exactly is the conformal type? Does it make sense to think of (closed) geodesics in this context?
$endgroup$
– Hans Stricker
Oct 23 '12 at 14:14
add a comment |
9
$begingroup$
This sounds suspiciously like the Klein bottle.
$endgroup$
– Fredrik Meyer
Oct 20 '12 at 17:49
$begingroup$
Are you looking for the Klein Bottle, which joins the ends of a cylinder in the opposite orientation to a torus?
$endgroup$
– Mark Bennet
Oct 20 '12 at 17:51
4
$begingroup$
A mere twist before identifying the ends of the "hose" again produces an ordinary torus with a locally euclidean metric, but its global conformal type has changed. If you identify the ends reversing the orientation you get a Klein bottle, which is a nonorientable surface.
$endgroup$
– Christian Blatter
Oct 20 '12 at 18:21
1
$begingroup$
Just to add to Christian's comment, a twist will be isotopic to the identity map, and so the resulting space will be homeomorphic to a regular torus. Think about it: if you can do it in R^3, it's not a Klein bottle. You need a degree -1 map on the circle boundaries to get the Klein bottle.
$endgroup$
– user641
Oct 21 '12 at 4:00
$begingroup$
@Christian Blatter: Can you please help me to understand better: the simply twisted torus (which I tried to ask for) has a locally euclidean metric (just like the Klein bottle), is homeomorphic to the regular torus, but has another conformal type. What exactly is the conformal type? Does it make sense to think of (closed) geodesics in this context?
$endgroup$
– Hans Stricker
Oct 23 '12 at 14:14
9
9
$begingroup$
This sounds suspiciously like the Klein bottle.
$endgroup$
– Fredrik Meyer
Oct 20 '12 at 17:49
$begingroup$
This sounds suspiciously like the Klein bottle.
$endgroup$
– Fredrik Meyer
Oct 20 '12 at 17:49
$begingroup$
Are you looking for the Klein Bottle, which joins the ends of a cylinder in the opposite orientation to a torus?
$endgroup$
– Mark Bennet
Oct 20 '12 at 17:51
$begingroup$
Are you looking for the Klein Bottle, which joins the ends of a cylinder in the opposite orientation to a torus?
$endgroup$
– Mark Bennet
Oct 20 '12 at 17:51
4
4
$begingroup$
A mere twist before identifying the ends of the "hose" again produces an ordinary torus with a locally euclidean metric, but its global conformal type has changed. If you identify the ends reversing the orientation you get a Klein bottle, which is a nonorientable surface.
$endgroup$
– Christian Blatter
Oct 20 '12 at 18:21
$begingroup$
A mere twist before identifying the ends of the "hose" again produces an ordinary torus with a locally euclidean metric, but its global conformal type has changed. If you identify the ends reversing the orientation you get a Klein bottle, which is a nonorientable surface.
$endgroup$
– Christian Blatter
Oct 20 '12 at 18:21
1
1
$begingroup$
Just to add to Christian's comment, a twist will be isotopic to the identity map, and so the resulting space will be homeomorphic to a regular torus. Think about it: if you can do it in R^3, it's not a Klein bottle. You need a degree -1 map on the circle boundaries to get the Klein bottle.
$endgroup$
– user641
Oct 21 '12 at 4:00
$begingroup$
Just to add to Christian's comment, a twist will be isotopic to the identity map, and so the resulting space will be homeomorphic to a regular torus. Think about it: if you can do it in R^3, it's not a Klein bottle. You need a degree -1 map on the circle boundaries to get the Klein bottle.
$endgroup$
– user641
Oct 21 '12 at 4:00
$begingroup$
@Christian Blatter: Can you please help me to understand better: the simply twisted torus (which I tried to ask for) has a locally euclidean metric (just like the Klein bottle), is homeomorphic to the regular torus, but has another conformal type. What exactly is the conformal type? Does it make sense to think of (closed) geodesics in this context?
$endgroup$
– Hans Stricker
Oct 23 '12 at 14:14
$begingroup$
@Christian Blatter: Can you please help me to understand better: the simply twisted torus (which I tried to ask for) has a locally euclidean metric (just like the Klein bottle), is homeomorphic to the regular torus, but has another conformal type. What exactly is the conformal type? Does it make sense to think of (closed) geodesics in this context?
$endgroup$
– Hans Stricker
Oct 23 '12 at 14:14
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
As remarked above: What you get is the Klein bottle. Put differently: The result is what you get when you take two Möbius strips (which both have one boundary) and glue both boundaries together (which does not work when embedded in 3d space but works in theory). See this image from http://im-possible.info/english/articles/klein-bottle/:
edited Dec 23 '18 at 20:27
Glorfindel
3,41981830
answered Oct 20 '12 at 18:09
DirkDirk
8,8002447
$endgroup$
2
$begingroup$
They might not meet all the technical details of the definition; but as long as you're willing to overlook the defect where it punches through its side, you can buy glass klein bottles as conversation pieces even if you're trapped in 3-space.
$endgroup$
– Dan Neely
Oct 20 '12 at 20:51
add a comment |
$begingroup$
In the course of experimenting with 3-D models, I developed a method for twisting a toroid of arbitrary cross section. I called these forms Möbioids. Of course, if the cross-section is circular the result will be indistinguishable. What I did was to consider cylinders of non-circular cross-section. However, then the twist angles are quantized. Depending on the particular cross-section and number of twists, you can get forms with one or more surfaces. The figures below show an astroid cross-section with one twist of $pi/4$ (left) and a pentacuspid cross-section with six twists of $2pi/5$ (right). Each has a single surface. Again, not all twists lead to a single surface. You can find more images and some animations at A New Twist on Möbius.
answered Apr 5 '17 at 16:49
Cye WaldmanCye Waldman
4,1252523
$endgroup$
add a comment |
$begingroup$
The question is one of boundary between two and three dimensional space. For example, bisect cut a bicycle tube which lays flat, draw a green line down one side of the tube, draw a red line down the other side of the tube. Then rotate one end 180-degrees and cement it to the other end. The green line meets the red line on both sides of the tube. It looks nothing like a Klein bottle. If you now inflate the tube from a two dimensional flat plane to a three dimensional torus, in effect you are creating an infinite loop where say, an ant travelling in one direction along the inside of the tube has twice the distance to walk to return to his original starting point. This topography is not found in nature, at least not on the surface of a sphere. If the tube is blown up enough, say to the size of a universe, you have finite space WITHIN the tube, although it is quite possible to have infinite space on the outside of the tube, surrounding it. We simply don't know at this point since all we can comprehend is our interior space. All that we do know is that we exist in a two-dimensional closed (finite) space within a three-dimensional open (infinite) space. To say we live in a three dimensional world (height, length, depth) is naive at best.
answered Jun 10 '14 at 15:49
Dr. Jerome MyerDr. Jerome Myer
1
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As remarked above: What you get is the Klein bottle. Put differently: The result is what you get when you take two Möbius strips (which both have one boundary) and glue both boundaries together (which does not work when embedded in 3d space but works in theory). See this image from http://im-possible.info/english/articles/klein-bottle/:
edited Dec 23 '18 at 20:27
Glorfindel
3,41981830
answered Oct 20 '12 at 18:09
DirkDirk
8,8002447
$endgroup$
2
$begingroup$
They might not meet all the technical details of the definition; but as long as you're willing to overlook the defect where it punches through its side, you can buy glass klein bottles as conversation pieces even if you're trapped in 3-space.
$endgroup$
– Dan Neely
Oct 20 '12 at 20:51
add a comment |
$begingroup$
As remarked above: What you get is the Klein bottle. Put differently: The result is what you get when you take two Möbius strips (which both have one boundary) and glue both boundaries together (which does not work when embedded in 3d space but works in theory). See this image from http://im-possible.info/english/articles/klein-bottle/:
edited Dec 23 '18 at 20:27
Glorfindel
3,41981830
answered Oct 20 '12 at 18:09
DirkDirk
8,8002447
$endgroup$
2
$begingroup$
They might not meet all the technical details of the definition; but as long as you're willing to overlook the defect where it punches through its side, you can buy glass klein bottles as conversation pieces even if you're trapped in 3-space.
$endgroup$
– Dan Neely
Oct 20 '12 at 20:51
add a comment |
$begingroup$
As remarked above: What you get is the Klein bottle. Put differently: The result is what you get when you take two Möbius strips (which both have one boundary) and glue both boundaries together (which does not work when embedded in 3d space but works in theory). See this image from http://im-possible.info/english/articles/klein-bottle/:
edited Dec 23 '18 at 20:27
Glorfindel
3,41981830
answered Oct 20 '12 at 18:09
DirkDirk
8,8002447
$endgroup$
As remarked above: What you get is the Klein bottle. Put differently: The result is what you get when you take two Möbius strips (which both have one boundary) and glue both boundaries together (which does not work when embedded in 3d space but works in theory). See this image from http://im-possible.info/english/articles/klein-bottle/:
edited Dec 23 '18 at 20:27
Glorfindel
3,41981830
answered Oct 20 '12 at 18:09
DirkDirk
8,8002447
edited Dec 23 '18 at 20:27
Glorfindel
3,41981830
edited Dec 23 '18 at 20:27
Glorfindel
3,41981830
edited Dec 23 '18 at 20:27
Glorfindel
3,41981830
3,41981830
answered Oct 20 '12 at 18:09
DirkDirk
8,8002447
answered Oct 20 '12 at 18:09
DirkDirk
8,8002447
answered Oct 20 '12 at 18:09
DirkDirk
8,8002447
8,8002447
2
$begingroup$
They might not meet all the technical details of the definition; but as long as you're willing to overlook the defect where it punches through its side, you can buy glass klein bottles as conversation pieces even if you're trapped in 3-space.
$endgroup$
– Dan Neely
Oct 20 '12 at 20:51
add a comment |
2
$begingroup$
They might not meet all the technical details of the definition; but as long as you're willing to overlook the defect where it punches through its side, you can buy glass klein bottles as conversation pieces even if you're trapped in 3-space.
$endgroup$
– Dan Neely
Oct 20 '12 at 20:51
2
2
$begingroup$
They might not meet all the technical details of the definition; but as long as you're willing to overlook the defect where it punches through its side, you can buy glass klein bottles as conversation pieces even if you're trapped in 3-space.
$endgroup$
– Dan Neely
Oct 20 '12 at 20:51
$begingroup$
They might not meet all the technical details of the definition; but as long as you're willing to overlook the defect where it punches through its side, you can buy glass klein bottles as conversation pieces even if you're trapped in 3-space.
$endgroup$
– Dan Neely
Oct 20 '12 at 20:51
add a comment |
$begingroup$
In the course of experimenting with 3-D models, I developed a method for twisting a toroid of arbitrary cross section. I called these forms Möbioids. Of course, if the cross-section is circular the result will be indistinguishable. What I did was to consider cylinders of non-circular cross-section. However, then the twist angles are quantized. Depending on the particular cross-section and number of twists, you can get forms with one or more surfaces. The figures below show an astroid cross-section with one twist of $pi/4$ (left) and a pentacuspid cross-section with six twists of $2pi/5$ (right). Each has a single surface. Again, not all twists lead to a single surface. You can find more images and some animations at A New Twist on Möbius.
answered Apr 5 '17 at 16:49
Cye WaldmanCye Waldman
4,1252523
$endgroup$
add a comment |
$begingroup$
In the course of experimenting with 3-D models, I developed a method for twisting a toroid of arbitrary cross section. I called these forms Möbioids. Of course, if the cross-section is circular the result will be indistinguishable. What I did was to consider cylinders of non-circular cross-section. However, then the twist angles are quantized. Depending on the particular cross-section and number of twists, you can get forms with one or more surfaces. The figures below show an astroid cross-section with one twist of $pi/4$ (left) and a pentacuspid cross-section with six twists of $2pi/5$ (right). Each has a single surface. Again, not all twists lead to a single surface. You can find more images and some animations at A New Twist on Möbius.
answered Apr 5 '17 at 16:49
Cye WaldmanCye Waldman
4,1252523
$endgroup$
add a comment |
$begingroup$
In the course of experimenting with 3-D models, I developed a method for twisting a toroid of arbitrary cross section. I called these forms Möbioids. Of course, if the cross-section is circular the result will be indistinguishable. What I did was to consider cylinders of non-circular cross-section. However, then the twist angles are quantized. Depending on the particular cross-section and number of twists, you can get forms with one or more surfaces. The figures below show an astroid cross-section with one twist of $pi/4$ (left) and a pentacuspid cross-section with six twists of $2pi/5$ (right). Each has a single surface. Again, not all twists lead to a single surface. You can find more images and some animations at A New Twist on Möbius.
answered Apr 5 '17 at 16:49
Cye WaldmanCye Waldman
4,1252523
$endgroup$
In the course of experimenting with 3-D models, I developed a method for twisting a toroid of arbitrary cross section. I called these forms Möbioids. Of course, if the cross-section is circular the result will be indistinguishable. What I did was to consider cylinders of non-circular cross-section. However, then the twist angles are quantized. Depending on the particular cross-section and number of twists, you can get forms with one or more surfaces. The figures below show an astroid cross-section with one twist of $pi/4$ (left) and a pentacuspid cross-section with six twists of $2pi/5$ (right). Each has a single surface. Again, not all twists lead to a single surface. You can find more images and some animations at A New Twist on Möbius.
answered Apr 5 '17 at 16:49
Cye WaldmanCye Waldman
4,1252523
answered Apr 5 '17 at 16:49
Cye WaldmanCye Waldman
4,1252523
answered Apr 5 '17 at 16:49
Cye WaldmanCye Waldman
4,1252523
answered Apr 5 '17 at 16:49
Cye WaldmanCye Waldman
4,1252523
4,1252523
add a comment |
add a comment |
$begingroup$
The question is one of boundary between two and three dimensional space. For example, bisect cut a bicycle tube which lays flat, draw a green line down one side of the tube, draw a red line down the other side of the tube. Then rotate one end 180-degrees and cement it to the other end. The green line meets the red line on both sides of the tube. It looks nothing like a Klein bottle. If you now inflate the tube from a two dimensional flat plane to a three dimensional torus, in effect you are creating an infinite loop where say, an ant travelling in one direction along the inside of the tube has twice the distance to walk to return to his original starting point. This topography is not found in nature, at least not on the surface of a sphere. If the tube is blown up enough, say to the size of a universe, you have finite space WITHIN the tube, although it is quite possible to have infinite space on the outside of the tube, surrounding it. We simply don't know at this point since all we can comprehend is our interior space. All that we do know is that we exist in a two-dimensional closed (finite) space within a three-dimensional open (infinite) space. To say we live in a three dimensional world (height, length, depth) is naive at best.
answered Jun 10 '14 at 15:49
Dr. Jerome MyerDr. Jerome Myer
1
$endgroup$
add a comment |
$begingroup$
The question is one of boundary between two and three dimensional space. For example, bisect cut a bicycle tube which lays flat, draw a green line down one side of the tube, draw a red line down the other side of the tube. Then rotate one end 180-degrees and cement it to the other end. The green line meets the red line on both sides of the tube. It looks nothing like a Klein bottle. If you now inflate the tube from a two dimensional flat plane to a three dimensional torus, in effect you are creating an infinite loop where say, an ant travelling in one direction along the inside of the tube has twice the distance to walk to return to his original starting point. This topography is not found in nature, at least not on the surface of a sphere. If the tube is blown up enough, say to the size of a universe, you have finite space WITHIN the tube, although it is quite possible to have infinite space on the outside of the tube, surrounding it. We simply don't know at this point since all we can comprehend is our interior space. All that we do know is that we exist in a two-dimensional closed (finite) space within a three-dimensional open (infinite) space. To say we live in a three dimensional world (height, length, depth) is naive at best.
answered Jun 10 '14 at 15:49
Dr. Jerome MyerDr. Jerome Myer
1
$endgroup$
add a comment |
$begingroup$
The question is one of boundary between two and three dimensional space. For example, bisect cut a bicycle tube which lays flat, draw a green line down one side of the tube, draw a red line down the other side of the tube. Then rotate one end 180-degrees and cement it to the other end. The green line meets the red line on both sides of the tube. It looks nothing like a Klein bottle. If you now inflate the tube from a two dimensional flat plane to a three dimensional torus, in effect you are creating an infinite loop where say, an ant travelling in one direction along the inside of the tube has twice the distance to walk to return to his original starting point. This topography is not found in nature, at least not on the surface of a sphere. If the tube is blown up enough, say to the size of a universe, you have finite space WITHIN the tube, although it is quite possible to have infinite space on the outside of the tube, surrounding it. We simply don't know at this point since all we can comprehend is our interior space. All that we do know is that we exist in a two-dimensional closed (finite) space within a three-dimensional open (infinite) space. To say we live in a three dimensional world (height, length, depth) is naive at best.
answered Jun 10 '14 at 15:49
Dr. Jerome MyerDr. Jerome Myer
1
$endgroup$
The question is one of boundary between two and three dimensional space. For example, bisect cut a bicycle tube which lays flat, draw a green line down one side of the tube, draw a red line down the other side of the tube. Then rotate one end 180-degrees and cement it to the other end. The green line meets the red line on both sides of the tube. It looks nothing like a Klein bottle. If you now inflate the tube from a two dimensional flat plane to a three dimensional torus, in effect you are creating an infinite loop where say, an ant travelling in one direction along the inside of the tube has twice the distance to walk to return to his original starting point. This topography is not found in nature, at least not on the surface of a sphere. If the tube is blown up enough, say to the size of a universe, you have finite space WITHIN the tube, although it is quite possible to have infinite space on the outside of the tube, surrounding it. We simply don't know at this point since all we can comprehend is our interior space. All that we do know is that we exist in a two-dimensional closed (finite) space within a three-dimensional open (infinite) space. To say we live in a three dimensional world (height, length, depth) is naive at best.
answered Jun 10 '14 at 15:49
Dr. Jerome MyerDr. Jerome Myer
1
answered Jun 10 '14 at 15:49
Dr. Jerome MyerDr. Jerome Myer
1
answered Jun 10 '14 at 15:49
Dr. Jerome MyerDr. Jerome Myer
1
answered Jun 10 '14 at 15:49
Dr. Jerome MyerDr. Jerome Myer
1
1
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This sounds suspiciously like the Klein bottle.
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– Fredrik Meyer
Oct 20 '12 at 17:49
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Are you looking for the Klein Bottle, which joins the ends of a cylinder in the opposite orientation to a torus?
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– Mark Bennet
Oct 20 '12 at 17:51
4
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A mere twist before identifying the ends of the "hose" again produces an ordinary torus with a locally euclidean metric, but its global conformal type has changed. If you identify the ends reversing the orientation you get a Klein bottle, which is a nonorientable surface.
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– Christian Blatter
Oct 20 '12 at 18:21
1
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Just to add to Christian's comment, a twist will be isotopic to the identity map, and so the resulting space will be homeomorphic to a regular torus. Think about it: if you can do it in R^3, it's not a Klein bottle. You need a degree -1 map on the circle boundaries to get the Klein bottle.
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– user641
Oct 21 '12 at 4:00
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@Christian Blatter: Can you please help me to understand better: the simply twisted torus (which I tried to ask for) has a locally euclidean metric (just like the Klein bottle), is homeomorphic to the regular torus, but has another conformal type. What exactly is the conformal type? Does it make sense to think of (closed) geodesics in this context?
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– Hans Stricker
Oct 23 '12 at 14:14