Problem on Do Carmo's definition of manifold
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In the book Riemannian Geometry, Do Carmo gave his definition of manifold without metionning to the topology on the set M. However, the usual way to define manifold is always to supposing M is a topological space with some restrictions on the topology, such as $A_{2}$ or $T_{2}$.
Later, he made a remark on this. As he said, we can induce a natural topology on M using the differential structure.
My question is that:
What happened if the topology induced in this way may not be $T_{2}$ or $A_{2}$? Do Carmo metioned this problem but he didn't give an answer. I am in a mess now...Help, thanks in advance.
differential-geometry manifolds riemannian-geometry smooth-manifolds
asked Nov 16 at 14:31
user450201
658
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In the book Riemannian Geometry, Do Carmo gave his definition of manifold without metionning to the topology on the set M. However, the usual way to define manifold is always to supposing M is a topological space with some restrictions on the topology, such as $A_{2}$ or $T_{2}$.
Later, he made a remark on this. As he said, we can induce a natural topology on M using the differential structure.
My question is that:
What happened if the topology induced in this way may not be $T_{2}$ or $A_{2}$? Do Carmo metioned this problem but he didn't give an answer. I am in a mess now...Help, thanks in advance.
differential-geometry manifolds riemannian-geometry smooth-manifolds
asked Nov 16 at 14:31
user450201
658
After introducing these axioms, does he make use of them later? They are both necessary, for example the line with doubled origin is permitted by his definition of differentiable manifold, but is not Hausdorff.
– Joppy
Nov 17 at 1:48
@Joppy He then introduced the partition of unity, which needs the condition of A_2. However, he didn’t say how to put this condition to the so called natural topology.
– user450201
Nov 17 at 1:56
I think the point is that in addition to his definition, you need to assume both these axioms in order to do differential geometry (such as using partitions of unity). These axioms do not follow from the definition.
– Joppy
Nov 17 at 1:57
@Joppy Yeah, I try to understand what he said with more modern definition. But that was quite a mess...Is there anyone reading the book before find this problem.? How does them get rid of it?
– user450201
Nov 17 at 2:02
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In the book Riemannian Geometry, Do Carmo gave his definition of manifold without metionning to the topology on the set M. However, the usual way to define manifold is always to supposing M is a topological space with some restrictions on the topology, such as $A_{2}$ or $T_{2}$.
Later, he made a remark on this. As he said, we can induce a natural topology on M using the differential structure.
My question is that:
What happened if the topology induced in this way may not be $T_{2}$ or $A_{2}$? Do Carmo metioned this problem but he didn't give an answer. I am in a mess now...Help, thanks in advance.
differential-geometry manifolds riemannian-geometry smooth-manifolds
asked Nov 16 at 14:31
user450201
658
In the book Riemannian Geometry, Do Carmo gave his definition of manifold without metionning to the topology on the set M. However, the usual way to define manifold is always to supposing M is a topological space with some restrictions on the topology, such as $A_{2}$ or $T_{2}$.
Later, he made a remark on this. As he said, we can induce a natural topology on M using the differential structure.
My question is that:
What happened if the topology induced in this way may not be $T_{2}$ or $A_{2}$? Do Carmo metioned this problem but he didn't give an answer. I am in a mess now...Help, thanks in advance.
differential-geometry manifolds riemannian-geometry smooth-manifolds
differential-geometry manifolds riemannian-geometry smooth-manifolds
asked Nov 16 at 14:31
user450201
658
asked Nov 16 at 14:31
user450201
658
edited Nov 17 at 4:49
asked Nov 16 at 14:31
user450201
658
asked Nov 16 at 14:31
user450201
658
asked Nov 16 at 14:31
user450201
658
658
After introducing these axioms, does he make use of them later? They are both necessary, for example the line with doubled origin is permitted by his definition of differentiable manifold, but is not Hausdorff.
– Joppy
Nov 17 at 1:48
@Joppy He then introduced the partition of unity, which needs the condition of A_2. However, he didn’t say how to put this condition to the so called natural topology.
– user450201
Nov 17 at 1:56
I think the point is that in addition to his definition, you need to assume both these axioms in order to do differential geometry (such as using partitions of unity). These axioms do not follow from the definition.
– Joppy
Nov 17 at 1:57
@Joppy Yeah, I try to understand what he said with more modern definition. But that was quite a mess...Is there anyone reading the book before find this problem.? How does them get rid of it?
– user450201
Nov 17 at 2:02
add a comment |
After introducing these axioms, does he make use of them later? They are both necessary, for example the line with doubled origin is permitted by his definition of differentiable manifold, but is not Hausdorff.
– Joppy
Nov 17 at 1:48
@Joppy He then introduced the partition of unity, which needs the condition of A_2. However, he didn’t say how to put this condition to the so called natural topology.
– user450201
Nov 17 at 1:56
I think the point is that in addition to his definition, you need to assume both these axioms in order to do differential geometry (such as using partitions of unity). These axioms do not follow from the definition.
– Joppy
Nov 17 at 1:57
@Joppy Yeah, I try to understand what he said with more modern definition. But that was quite a mess...Is there anyone reading the book before find this problem.? How does them get rid of it?
– user450201
Nov 17 at 2:02
After introducing these axioms, does he make use of them later? They are both necessary, for example the line with doubled origin is permitted by his definition of differentiable manifold, but is not Hausdorff.
– Joppy
Nov 17 at 1:48
After introducing these axioms, does he make use of them later? They are both necessary, for example the line with doubled origin is permitted by his definition of differentiable manifold, but is not Hausdorff.
– Joppy
Nov 17 at 1:48
@Joppy He then introduced the partition of unity, which needs the condition of A_2. However, he didn’t say how to put this condition to the so called natural topology.
– user450201
Nov 17 at 1:56
@Joppy He then introduced the partition of unity, which needs the condition of A_2. However, he didn’t say how to put this condition to the so called natural topology.
– user450201
Nov 17 at 1:56
I think the point is that in addition to his definition, you need to assume both these axioms in order to do differential geometry (such as using partitions of unity). These axioms do not follow from the definition.
– Joppy
Nov 17 at 1:57
I think the point is that in addition to his definition, you need to assume both these axioms in order to do differential geometry (such as using partitions of unity). These axioms do not follow from the definition.
– Joppy
Nov 17 at 1:57
@Joppy Yeah, I try to understand what he said with more modern definition. But that was quite a mess...Is there anyone reading the book before find this problem.? How does them get rid of it?
– user450201
Nov 17 at 2:02
@Joppy Yeah, I try to understand what he said with more modern definition. But that was quite a mess...Is there anyone reading the book before find this problem.? How does them get rid of it?
– user450201
Nov 17 at 2:02
add a comment |
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After introducing these axioms, does he make use of them later? They are both necessary, for example the line with doubled origin is permitted by his definition of differentiable manifold, but is not Hausdorff.
– Joppy
Nov 17 at 1:48
@Joppy He then introduced the partition of unity, which needs the condition of A_2. However, he didn’t say how to put this condition to the so called natural topology.
– user450201
Nov 17 at 1:56
I think the point is that in addition to his definition, you need to assume both these axioms in order to do differential geometry (such as using partitions of unity). These axioms do not follow from the definition.
– Joppy
Nov 17 at 1:57
@Joppy Yeah, I try to understand what he said with more modern definition. But that was quite a mess...Is there anyone reading the book before find this problem.? How does them get rid of it?
– user450201
Nov 17 at 2:02