What is...the Gauss-Bonnet theorem?

[@Tanayatripathy1235]

This was really helpful, thank you so much😌

[@VisualMath]

Welcome, I am glad that you found the video helpful. That is why I do it - and also because its fun (very selfish, shameful....;-) )

[@kangmoabel]

In short gauss bonnet theroem = 2pi * ( V - E + F)

[@VisualMath]

Hah, I like that! Thank you for sharing. Maybe even shorter: Gauss-Bonnet = 2pi * Euler ;-)

[@pixplays5007]

i dont understand why the inside of the torus has negative curvature

[@VisualMath]

In the usual parametrization (you need to choose coordinates) of the torus, the sign of the curvature is determined by the angle of the meridian. To see that is a nontrivial but also not hard calculation. Instead of trying to do it here in the comments, which will be a mess, let me rather link a nice explanation: www.rdrop.com/~half/math/torus/curvature.xhtml I hope that helps!

[@justanormalyoutubeuser3868]

The curvature is negative if there are to cross-sections that curve in opposite ways.