7

On your more general question about differential geometry, i.e. why do people study it? There are many answers, some having little to do with each other. In my opinion differential geometry is perhaps best approached once you have seen the need for formal and computational structure underlying various fairly mundane geometric problems in more than one area....


7

If you are interested in local-to-global results, i.e., collecting local info about the manifold and then patch it together to get a global info then you need tools for the patching part of the process. These often come under the guise of some form of integration. Here is a simple example. If a compact surface admits a metric with negative curvature (local ...


4

This might be my own bias, but I think differential geometry is a really natural area to study. When we look out at the world around us, we see lots of objects that seem smooth, but are not flat. As such, it is only natural to try to study geometry using techniques from calculus, which is the starting point of differential geometry. In this vein, many of the ...


3

I don’t understand modern abstract differential geometry, but the elementary theory of curves and surfaces in $\mathbb{R}^3$, as expounded by Gauss, Euler, Darboux and others, is very useful in engineering and manufacturing. A few examples: The curvature properties of a surface determine how light reflects from it. These reflections are what determine the ...


3

I would like to know why you are interested in the specific $d$-dimensional measures given by the Hausdorff ones. For $d=n$ such a choice is understandable since the Lebesgue measure in $\mathbb{R}^{n}$ is proportional to $\mathcal{H}^{n}$. But (to the best of my knowledge) there is no nice description in the case of $d$-dimensional Hausdorff measures. A ...


2

The real motivation for the Lie derivative is doing differential calculus with vector fields. If we want fo differentiate the vector field $W$ in the direction of the vector field $V$, we take the flow of $V$ through time, use it to pull back $W$, and take the derivative at $t=0$. To explain better, let $V$ and $W$ be vector fields on a smooth manifold $M$, ...


1

Note that $f$ is smooth as a map of varieties. If $f$ comes from a direct product $\mathbb{C}^3\approx S\times \mathbb{C}$ then $S\approx \mathbb{C}^2$. See Section 5.1 of $\mathbb{A}^1$-homotopy theory and contractible varieties: a survey I do not see why $f$ must come from a direct product so this is not a complete answer.


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