What is differential geometry and why do we care about it?

In a nutshell, the main idea in differential geometry is the following: we want to study curved spaces of a certain dimension, taking advantage of many useful tools from calculus.
Okay, great but…

1- What kind of spaces?
2- What does it mean “certain dimension”?
3- Why would we need calculus to study geometry?

Today I’ll try to give an answer to these questions.

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Tags: Geometry