- #1
joda80
- 18
- 0
Dear All,
I've been studying differential geometry for some time, but there is one thing I keep failing to understand. Perhaps you can help out (I think the question is quite simple):
Can I use Cartesian coordinates to cover a curved manifold? I.e., is there an atlas that only contains Cartesian coordinate patches (I envision small squares covering a sphere, which should work if the squares overlap).
On the one hand I think this should be possible as the charts are allowed (and indeed, supposed) to overlap.
On the other hand, in classical diff.-geo. textbooks (not using manifolds), it says that no Cartesian coordinates are possible on curved manifolds.
How can these two ideas be reconciled?
Thanks a lot and best regards,
Johannes
I've been studying differential geometry for some time, but there is one thing I keep failing to understand. Perhaps you can help out (I think the question is quite simple):
Can I use Cartesian coordinates to cover a curved manifold? I.e., is there an atlas that only contains Cartesian coordinate patches (I envision small squares covering a sphere, which should work if the squares overlap).
On the one hand I think this should be possible as the charts are allowed (and indeed, supposed) to overlap.
On the other hand, in classical diff.-geo. textbooks (not using manifolds), it says that no Cartesian coordinates are possible on curved manifolds.
How can these two ideas be reconciled?
Thanks a lot and best regards,
Johannes