Copyright (C) 2018-2021 YuqiaoZhang(HanetakaChou)
This program is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public License along with this program. If not, see <https://www.gnu.org/licenses/>
I am a mathematics amateur and I am learning the book "Visual Differential Geometry and Forms" in my leisure time.
Since I am not professional, I am publishing my solutions here and I am willing to accept your criticism. If you find there are any mistakes or improvements, please don't hesitate to contact me by email.
The ultimate equality notation "≍" actually represents the equivalent infinitesimal which can be checked by the Taylor's theorem.
The most important point of the example of the negative curvature is to wave up and down simultaneously.
By (1.3), we have
And since "ε ≡ (angle sum of triangle) − π", we have