Torus - Singh, Example 2.2.5 - Baffled by certain aspects

[Math Amateur]

I am baffled by some aspects of the torus ... I hope someone can help ...

I am puzzled by some aspects of Singh's treatment of the torus in Example 2.2.5 ( Tej Bahadur Singh: Elements of Topology, CRC Press, 2013) ... ...

Singh's Example 2.2.5 reads as follows:

My questions related to the above example of Singh's are as follows:Question 1

$C_1$ as defined above seems to me to be a circle at 'height' $z = 1$, around the $z$-axis ... why (for what reason?) did Singh choose $C_1$ to be at 'height' $z = 1$?

Why not choose $C_1$ as $\{ (x,y, 0) \ | \ x^2 + y^2 = 4 \}$?

Such a choice seems more natural if you are rotating $C_2$ around the $z$-axis, since $C_1$ is at level $z = 0$ ... ...Question 2

$h \ : \ T \rightarrow C_1 \times C_2$

maps $(x,y,z)$ onto two three dimensional points in Euclidean $3$-space and so essentially maps $(x,y,z)$ into Euclidean $6$-space ... ..

... BUT ...

$T$ is homeomorphic to $S^1 \times S^1$ which is embedded in Euclidean $4$-space ... ... how can this be ...


I hope someone can clarify the above issues/questions ...

Peter

 

[lavinia]

Question 2: C1 and C2 are both circles in R^3 so the product is embedded in R^6.

Question 1. The top of the torus,T, is a circle at height of 1 above the xy-plane. and has radius 2
The circle C2 is the same circle.

 

[Math Amateur]

I am baffled by some aspects of the torus ... I hope someone can help ...

I am puzzled by some aspects of Singh's treatment of the torus in Example 2.2.5 ( Tej Bahadur Singh: Elements of Topology, CRC Press, 2013) ... ...

Singh's Example 2.2.5 reads as follows:

My questions related to the above example of Singh's are as follows:Question 1

$C_1$ as defined above seems to me to be a circle at 'height' $z = 1$, around the $z$-axis ... why (for what reason?) did Singh choose $C_1$ to be at 'height' $z = 1$?

Why not choose $C_1$ as $\{ (x,y, 0) \ | \ x^2 + y^2 = 4 \}$?

Such a choice seems more natural if you are rotating $C_2$ around the $z$-axis, since $C_1$ is at level $z = 0$ ... ...Question 2

$h \ : \ T \rightarrow C_1 \times C_2$

maps $(x,y,z)$ onto two three dimensional points in Euclidean $3$-space and so essentially maps $(x,y,z)$ into Euclidean $6$-space ... ..

... BUT ...

$T$ is homeomorphic to $S^1 \times S^1$ which is embedded in Euclidean $4$-space ... ... how can this be ...


I hope someone can clarify the above issues/questions ...

Peter

Question 2: C1 and C2 are both circles in R^3 so the product is embedded in R^6.

Question 1. The top of the torus,T, is a circle at height of 1 above the xy-plane. and has radius 2
The circle C2 is the same circle.

Thanks so much for your help Lavinia ... ...

... ... BUT ... for Question 1 ... do you mean "The circle C1 is the same circle"and ... if it is supposed to be C_1, then could Singh have alternatively, defined C_1 as

$C_1$ as $\{ (x,y, -1) \ | \ x^2 + y^2 = 4 \}$ ...

as this seems to define the bottom of the torus ...

Hope you can help further ... ...

Peter

 

[lavinia]

Thanks so much for your help Lavinia ... ...

... ... BUT ... for Question 1 ... do you mean "The circle C1 is the same circle"and ... if it is supposed to be C_1, then could Singh have alternatively, defined C_1 as

$C_1$ as $\{ (x,y, -1) \ | \ x^2 + y^2 = 4 \}$ ...

as this seems to define the bottom of the torus ...

Hope you can help further ... ...

Peter

yes he could have defined C_1 as the bottom of the torus.

 

[Math Amateur]

Thanks Lavinia ... Appreciate the help ...

Just quickly ... In your answer to question 1, did you mean: The Circle C1 is the same circle?

Peter

 

[lavinia]

Thanks Lavinia ... Appreciate the help ...

Just quickly ... In your answer to question 1, did you mean: The Circle C1 is the same circle?

Peter

yes. sorry

 

[Math Amateur]

Thanks again Lavinia ... most helpful ...

Peter