I Parameterized surfaces from coordinates

127
28
For all parameterized (hyper)surfaces that form smooth manifolds of dimension ##n-1## embedded in Euclidean ##\mathbb {R}^n##, will there always exist a coordinate system ##\partial_{\bar \mu}## on ##\mathbb {R}^n## that yields the same manifold when the right coordinate (say ##\partial_1##) is set to the right constant such that the induced metric on the (sub)manifold is equal to ##g_{\bar \mu \bar \nu}## where any components that have a 1 are dropped?
 

Orodruin

Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
15,844
5,842
Locally, yes. Just take any coordinate system on the surface and use the distance from the surface as your final coordinate. That coordinate system will locally be a coordinate system in the full space.
 
127
28
Thanks!
 

Want to reply to this thread?

"Parameterized surfaces from coordinates" You must log in or register to reply here.

Related Threads for: Parameterized surfaces from coordinates

Replies
1
Views
7K
Replies
14
Views
1K
Replies
0
Views
2K
Replies
3
Views
2K
  • Posted
Replies
3
Views
4K
Replies
4
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top