Could someone suggest me a good book to start differential geometry which is not very hard to start with? I have learnt several variable calculus in the previous semester but haven't yet read any stuff about differential geometry.
I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.  I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right now. I must teach myself all the stuff by reading books.  Towards this purpose I want to know what are the most important basic theorems in differential geometry and differential topology. For a start, for differential topology, I think I must read Stokes' theorem and de Rham theorem with complete proofs.   Differential geometry is a bit more difficult. What is a connection? Which notion should I use? I want to know about parallel transport and holonomy. What are the most important and basic theorems here? Are there concise books which can teach me the stuff faster than the voluminous Spivak books?  Also finally I want to read into some algebraic geometry and Hodge/Kähler stuff.  Suggestions about important theorems and concepts to learn, and book references, will be most helpful.
Convolution operator has norm 1 with respect to $L^2$ norm Let $C_c(\mathbb{R})$ be the following:  $$C_c = \{ f \in C(\mathbb{R}) \mid \exists \text{ } T &gt; 0 \text{ s.t. } f(t) = 0  \text{ for } |t| \geq T\}$$  Let $T_n \in L(C_c(\mathbb{R}))$ be a linear operator such that:  $$T_n u = \delta_n *u, \forall u \in C_c(\mathbb{R}),$$  where  $$ \delta_n(t)= \begin{cases}        n^2(t+1/n) &amp; -1/n \leq t \leq 0 \\       -n^2(t-1/n) &amp; 0 &lt; t \leq 1/n \\       0 &amp; \text{elsewhere}     \end{cases} $$  I have to prove that with respect to the 2-norm, the operator $T_n$ has $\|T_n\| = 1$ and I have succeeded in proving that $\|T_nu\|_2 \leq \|u\|_2, \forall u \in C_c(\mathbb{R}).$  Now, I remained  to prove that $\exists \text{ } u \in C_c(\mathbb{R}) \text{ s.t. } \|T_nu\|_2 = \|u\|_2$, but I really don't get how $u$ should be shaped in order to satisfy it. During real, everyday use, that is?  I'm looking to upgrade a drive in a MacBook Pro, and am looking at either a 750GB/7200RPM or a 1TH/5400RPM drive.  Would I care about the speed difference? Is it worth giving up the additional space on the larger drive? Can I run CAT5/6 cables parallel to electrical cables? I am building a new house and am planning to hardwire network cables into each room.   Can I run CAT5/6 cables parallel to electrical wires without introducing any general safety issues or losing much in connection quality?  Wiring runs for up to 25 metres from patch panel and distribution board in garage to rooms.   My research indicates that is should be fine if they are in separate conduits or if the CAT cables are shielded. character "5" repeated in text fields, and system freezes I am running Ubuntu 15.10, with the latest updates.  When I type in a text field (google search, type name of file, etc..) the number 5 gets typed on its own and it keeps repeating until I press ESC and when I try to continue typing the number 5 resumes filling in the text field.  I have tried to set the keyboard repeat to zero, but does not help.  This does not happen all the time, sometimes if I shut down the computer it then behaves normally when I restart it, other times the computer becomes useless because I cannot type in any of the text fields.  Thanks Sal Atmospheric lighting in cycles How would I go a about making volume light beams?  I have have tried  various tutorials, but they must be for a older version. as applying them does seemingly nothing.  A link to a functioning tutorial would be greatly appreciated as well.  I don't have a picture or .blend to show, as my problem with atmospheric lighting does not pertain to any specific scene. I found a better proof for a theorem than a proof from a published paper. Is it possible for me to publish my proof with the same theorem citing that paper? Difference between cp -r and cp -R (copy command) Xerces error: org.apache.xerces.impl.dv.dtd.DTDDVFactoryImpl Same mesh with different colors How can I find tv shows/films similar to one I like? As a general case, how can I find movies or TV shows similar to an existing movie or TV show that I like? What is the best way to repeatedly execute a function every x seconds? I want to repeatedly execute a function in Python every 60 seconds forever (just like an  in Objective C). This code will run as a daemon and is effectively like calling the python script every minute using a cron, but without requiring that to be set up by the user.  In , the solution appears to effectively just  for x seconds. I don't need such advanced functionality so perhaps something like this would work  while True:     # Code executed here     time.sleep(60)   Are there any foreseeable problems with this code? How can I know what speed card to get for my camera? Abandoned colony of military engineer corps Gray cone obscuring/clipping the 3D Viewport When looking at my objects in 3d view I'm seeing a gray cone that seems to be hiding other objects.   Can anyone help me turn this off and explain what it is please.  Screenshots attached. Help evaluating a gamma function "Calling dibs" - what does it imply? The term "to have dibs on something" or "to call dibs on something" plays a recurring role in American film and television (e.g. ), so it gets exported a lot.   describes "dibs" as     [...] a common, "informal" convention to reserve or declare full or partial ownership of a community resource, such as a chair [...]   From the usages in the media I have seen, this convention also seems to contain some moral concept which is not described in dictionaries. For me, it looks like by calling dibs (or ), the caller actually receives the (implicit, moral) right to that resource, and this typically goes undisputed by their rivals.  Can you confirm the existence of this concept? Is this something that varies between groups of people (or maybe between regions), or is there a concensus on how to respond to this? I am really having trouble working a problem in my course. I will post it and then add what I have thought etc.  Let p be an odd prime, then using the Euclidian Algorithm, calculated the GCD of of $x^{p-1}-1$ and $x^2+1$ in $\mathbb{Z} / p\mathbb{Z}[x]$  Also, how does $p=1 (mod4)$ tell us that  $-1$ is a square in $\mathbb{Z} / p\mathbb{Z}$, infact that $-1$ is a square here if and only if $p=1(mod4)$.  My thoughts:   I know that $$x^{p}-x=(x)(x-1)(x-2)…(x-(p-1))$$ and so that in this sense our roots will be the congruence classes of $1,2,…p-1$  and also that $$x^{p-1}-1=(x-1)(x-2)…(x-(p-1))$$  But I get lost when trying to apply the EA to these polynomials,  like how can I do it when p is arbitrary?  Preferably id like to be able to understand/see how to do the first part first i.e. using division with residue and then move on to the second part. Any insight/hint or solution to this would be great,  Thank you all  Update:  When I try to do division with residue, I just get  $$x^{p-1}-1=(x^2+1)(x^{p-3}-x^{p-5}+x^{p-7}-x^{p-9}...)$$  for example,  $p=7$  $$(x^{6}-1)=(x^2+1)(x^4-x^2+1)$$  for $p=11$  $$(x^{10}-1)=(x^2+1)(x^{8}-x^{6}+x^{4}-x^{2}+1)$$  which for specific cases tells me that 1 is the gcd, but I dont know how to generalize that for the case with arbitrary prime p. Two spacecraft pilots travel at light speed, meet back on Earth in different era due to time dilation I've been looking for the name of a sci-fi romance book about space travel but haven't been successful so far. I came across it a few years ago, read the plot on its back cover but didn't read the book. It is about two spacecraft pilots, a man and a woman, who travelled at light speed (or maybe FTL, I can't remember) in separate ships. Every time they return to Earth, centuries have passed on Earth because of time dilation. They meet each other for a few times on Earth in different eras, but don't know when, if ever, they will meet again. I don't know how it ends.   If anyone knows which book I'm talking about, could you please tell me the name? Thanks a lot! Why does Java not have block-scoped variable declarations? Calculating square roots using the recurrence $x_{n+1} = \frac12 \left(x_n + \frac2{x_n}\right)$ Let $x_1 = 2$, and define $$x_{n+1} = \frac{1}{2} \left(x_n + \frac{2}{x_n}\right).$$ Show that $x_n^2$ is always greater than or equal to $2$, and then use this to prove that $x_n − x_{n+1} ≥ 0$. Conclude that $\lim x_n = \sqrt2$.   My question:  So I know how to do this problem but I don't know how to prove that $x_n &gt;0$ for all $n\in \mathbb{N}$.