Does a torus have positive curvature everywhere?
Answers
Question :-
Does a torus have positive curvature everywhere?
Answer :-
It is a theorem of Hilbert that any closed smooth surface without boundary in 3 space must have a point of positive Gauss curvature. So any surface that also has a point of negative Gauss curvature will have both.
The integral of the Gauss curvature over a surface is 2pi times its Euler characteristic. The torus has Euler characteristic zero so it must always have regions of negative Gauss curvature to cancel the regions of positive curvature guaranteed in Hilbert's theorem. There is no way to warp it so that its Gauss curvature has only one sign. Similarly surfaces with more than one handle all have negative Euler characteristic and thus must always have both positive and negative Gauss curvature.
This is not true for the torus in 4 space. Here the torus can be given zero Gauss curvature everywhere.
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
Learn more from brainly :
Why does the toru dutt consider the scene among the bamboos the loveliest spot in the garden? What effect does this beauty have on her.
https://brainly.in/question/20860355
Answer:
Yes
Step-by-step explanation:
The torus has Euler characteristic zero so it must always have regions of negative Gauss curvature to cancel the regions of positive curvature guaranteed in Hilbert's theorem. ... Here the torus can be given zero Gauss curvature everywhere.
...Thanks me later!!!