Register to reply 
Manifold definition 
Share this thread: 
#19
Feb913, 10:23 AM

Sci Advisor
HW Helper
P: 9,499

i assumed lavinia was thinking of smooth manifolds and thinking of the infinitesimal vector space, i.e. tangent space, structure that gives at a point.



#20
Feb913, 10:27 AM

PF Gold
P: 1,622




#21
Feb913, 02:01 PM

P: 3,060

As I see it this lack of a canonical basis for the local charts on the the manifold is precisely what makes the use of connections necessary to relate vectors from different tangent spaces linearizing open neighbourhoods at different points. 


#22
Feb1013, 09:10 AM

P: 3,060

I've been pondering a bit on these replies, at first they seemed to me quite reasonable, right now I feel they were not so on target.
so it seems the algebraic structure of vector spaces, namely linearity, has some important role in manifolds. 


#23
Feb1113, 01:28 AM

Mentor
P: 18,346

If we talk about topological manifolds, then the linear structure seems much less important. There is no (useful) analogue for tangent spaces. Furthermore, I rarely need a linear structure when talking about topological manifolds. 


#24
Feb1113, 02:45 AM

P: 3,060




#25
Feb1113, 04:25 AM

P: 3,060

I would like to add that even though what you are saying is true in general, in special cases that are heavily used in physics and engineering the distinction between topological and differentiable manifolds vanishes in the sense that for low dimensional manifolds(n<4) not only all diffeomorphisms are homeomorphisms like it's always the case, but also all homeomorphisms are diffeomorphisms, there is a unique differentiable structure so that all charts are smooth.



#26
Feb1113, 05:15 AM

Mentor
P: 18,346




#27
Feb1113, 05:17 AM

C. Spirit
Sci Advisor
Thanks
P: 5,661




#28
Feb1113, 05:36 AM

P: 3,060




#29
Feb1113, 05:39 AM

Mentor
P: 18,346




#30
Feb1113, 05:46 AM

P: 3,060

Gee, this is tough and I'm too fast drawing conclusions. I'll try to refrain that. 


#31
Feb1113, 09:40 AM

Sci Advisor
PF Gold
P: 1,594

People are actually trying to help you Tricky, best listen and ask for clarification where you don't understand, rather than jumping down their throats. 


#32
Feb1113, 01:06 PM

P: 3,060




#33
Feb1113, 02:10 PM

C. Spirit
Sci Advisor
Thanks
P: 5,661

Just be careful in what structures you pre  impose on your manifold. Things that hold for smooth manifolds don't necessarily have to hold for the more general topological manifold. The point is that in field theories like GR or classical field theory, you almost always only deal with smooth manifolds for obvious reasons so the mention of topological manifolds doesn't really come up but that certainly doesn't mean there is a vanishing of the distinction between topological manifolds and smooth manifolds.



#34
Feb1113, 03:12 PM

Sci Advisor
PF Gold
P: 1,594

I see in your subsequent responses you acted differently. In the past, you often got belligerent; maybe you're not anymore. 


#35
Feb1113, 03:52 PM

P: 3,060




#36
Feb1113, 04:05 PM

P: 3,060

And your tone is so patronizing, please quit it. Having said that I have always praised you as an expert in differential geometry in these forums so I encourage you to keep helping people around here. 


Register to reply 
Related Discussions  
Definition of manifold  Differential Geometry  4  
Definition of Manifold  Calculus  9  
Definition of a manifold  Differential Geometry  24  
Another manifold definition deficiency?  Differential Geometry  5  
Definition of differentiability on a manifold  Differential Geometry  15 