Intereting Posts

Axiom of Choice: What exactly is a choice, and when and why is it needed?
Proving identity $ \binom{n}{k} = (-1)^k \binom{k-n-1}{k} $. How to interpret factorials and binomial coefficients with negative integers.
Understanding the trivial primality test
Is there a systematic way to solve in $\bf Z$: $x_1^2+x_2^3+…+x_{n}^{n+1}=z^{n+2}$ for all $n$?
Does there exist any uncountable group , every proper subgroup of which is countable?
There are $12$ stations between A and B, in how many ways you can select 4 stations for a halt in such a way that no two stations are consecutive
Tensors as matrices vs. Tensors as multi-linear maps
Separating a point and a closed subset in a locally compact Hausdorff space
Evaluation of $\int_0^1 \frac{\log^2(1+x)}{x} \ dx$
What is a constructive proof of $\lnot\lnot(P\vee\lnot P)$?
What does this double sided arrow mean?
Evaluate the integral $\int_0^{\infty} \lfloor x \rfloor e^{-x}\mathrm dx$
“Basis extension theorem” for local smooth vector fields
Combinatoric proof for $\sum_{k=0}^n{n\choose k}\left(-1\right)^k\left(n-k\right)^4 = 0$ ($n\geqslant5$)
The concept of premeasure

I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know? I know some basic concepts reading from the internet on topological spaces, connectedness, compactness, metrics and quotient hausdorff spaces. Do I need to go deeper? Also, could you suggest me some chapters from topology textbooks to brush up this knowledge? Could you please also suggest a good differential geometry books that covers the topics in differential geometry that are needed in physics in sufficient detail (without too much emphasis on mathematical rigour)? I have heard of the following textbook authors: Nakhara, Fecko, Spivak. Would you recommend these?

- Brouwer transformation plane theorem
- finding sup and inf of $\{\frac{n+1}{n}, n\in \mathbb{N}\}$
- Ricci SCALAR curvature
- Simply connectedness in $R^3$ with a spherical hole?
- Geometric Interpretation: Parallel forms are harmonic
- Borel Measures: Atoms (Summary)
- Number of path components of a function space
- Urysohn's Lemma needn't hold in the absence of choice. Alternate terminology for inequivalent definitions of “normal” spaces?
- Example of a Borel set that is neither $F_\sigma$ nor $G_\delta$
- question related to perfect maps preserving compactness

You shouldn’t need much. Almost all you need to know about topology (especially of the point-set variety) should have been covered in a course in advanced calculus. That is to say, you really need to know about “stuff” in $\mathbb{R}^n$. (The one main exception is when you study instantons and some existence results are topological in nature; for that you will need to know a little bit about fundamental groups and homotopy.) The reason is that *differential topology* and *differential geometry* study objects which *locally* look like Euclidean spaces. This dramatically rules out lots of the more esoteric examples that point-set topologists and functional analysts like to consider. So most introductory books in differential geometry will quickly sketch some of the basic topological facts you will need to get going.

In terms of topology needed for differential geometry, one of the texts I highly recommend would be

- J.M. Lee’s
*Introduction to Smooth Manifolds*

It is quite mathematical and quite advanced, and covers large chunks of what you will call differential geometry also. One can complement that with his *Riemannian Manifolds* to get some Riemannian geometry also.

But since you are asking from the point of view of a Physics Undergrad, perhaps better for you would be to start with either (or both of)

- Nakahara’s
*Geometry, Topology, and Physics* - Choquet-Bruhat’s
*Analysis, Manifolds, and Physics*: Vol. 1 and Vol. 2

and follow-up with

- Greg Naber’s two book series:
*Topology, Geometry, and Gauge Fields: Foundations*and*Topology, Geometry, and Gauge Fields: Interactions*

- Integration double angle
- Examples of preorders in which meets and joins do not exist
- Applications of Principal Bundle Construction: Vague Question
- Automorphism group of the quaternion group
- Uniform Integrbility
- A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded
- Absolute continuity on an open interval of the real line?
- Is this “theorem” true in Optimization Theory?
- Evaluating $\lim\limits_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} – \frac{n}{p+1}\right)$
- $a-b,a^2-b^2,a^3-b^3…$ are integers $\implies$ $a,b$ are integers?
- Show that $d_b(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric.
- A curious equation containing an integral $\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}$
- Calculating CRC by long division: How to decide the top number of long division?
- Bernstein polynomial looks like this: $B_i^n={{n}\choose{i}}x^i(1-x)^{n-i}$.Find it's $r$'th derivative.
- study of subspace generated by $f_k(x)=f(x+k)$ with f continuous, bounded..