Changes

* Can the manifold be "flattened out"?  Any piece of spaghetti can be made straight, and yet an orange peel cannot be flattened without tearing.  How can we tell if it is possible to flatten a given surface?  This question leads to the study of ''curvature''.

* Can the manifold be "flattened out"?  Any piece of spaghetti can be made straight, and yet an orange peel cannot be flattened without tearing.  How can we tell if it is possible to flatten a given surface?  This question leads to the study of ''curvature''.

* Is there a natural notion of distance on the manifold?  In the case of a surface in R^3, the answer is yes: just define the length of a path in the surface to be the length of that path in R^3!

* Is there a natural notion of distance on the manifold?  In the case of a surface in R^3, the answer is yes: just define the length of a path in the surface to be the length of that path in R^3!

* How should one [[derivative|differentiate]] a function?  What does it mean to take the derivative of a function on the sphere?  This question leads to the development of objects called ''connections''.

* How should one [[Derivative (calculus)|differentiate]] a function?  What does it mean to take the derivative of a function on the sphere?  This question leads to the development of objects called ''connections''.

Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].  

Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].