Let $f:M\rightarrow N$ be a smooth map; surjective submersion and all that is necessary to call it a well behaved map.  Let $\Gamma$ be a connection on $N$ (a connection on the tangent bundle $TN\rightarrow N$).  Can I talk about the pull back of connection $\Gamma$ along $f:M\rightarrow N$ to get a connection on $M$?  I do not mean to ask for connection on the vector bundle obtained by pullback of vector bundle $TN\rightarrow N$ along the map $f:M\rightarrow N$ namely the vector bundle $M\times_N TN\rightarrow M$. No, I am not asking about connection on this vector bundle $M\times_NTN\rightarrow M$. I am asking for connection on the tangent bundle $TM\rightarrow M$.  Given a connection on tangent bundle $TN\rightarrow N$, can I get a connection on tangent bundle $TM\rightarrow M$ obtained by pull back of connection on $TN\rightarrow N$ along the map $f:M\rightarrow N$?  You can "twist" the notion of "pullback" slightly. You can assume $f$ is well behaved except that it is a diffeomorphism.
Suppose that $f:M \to N$ is a smooth map of two manifolds and $E$ is a vector bundle over $N$. Given a connecton $\nabla$ on $E$ one can pull it back to $f^*N$ (which is a bundle over $M$) and obtain a connection $f^* \nabla$.   If $E=TN$ is a tangent bundle one can consider the Levi-Civita connection $\nabla$, with respect to the chosen metric $g$ on $N$. It is characterised uniquely as torsion free metric connection. Given a map $f:M \to N$ one can pullback the metric $g$ to $f^*g$. If it happens that $f^*g$ is again a metric (which is the case for example for immersions) one can consider the Levi Civita connection on $M$ with respect to $f^*g$. I wonder how these two constructions are related, in particular      Is there a way to pullback affine connections to get again affine connection (i.e. connections on tangent bundle)?     Additionally I would like to know:     What is the relation between Christoffels symbols for the connection and for its pullback?
Why is `enjoy to [verb]` incorrect I can say:     I love to read. I hate to read.   But it must be:     I enjoy to read. I enjoy reading.   What is enjoy so different? Let's suppose I'm a wizard casting a ranged spell. An enemy is behind cover but I suspect he'll come out, so I ready an action to cast at him when he comes out; but I don't want to risk getting charged at by another enemy and losing concentration or taking disadvantage. Can I say:  "I ready a spell. When that guy over there leaves cover, or when any enemy gets within 5 feet of me, I will cast it on that enemy"? Quick doubt about multiplying vectors by scalars I've just started studying linear algebra, and I'm having trouble understanding something that apparentely seems obvious to everyone else. My question is, why does multiplying a vector by a positive scalar does not change its direction? I've seen some visual representations of this and it does seem reasonable to me that that should be the case. However, I wonder whether there is a way to actually show it mathematically? In the 2d coordinate plane, for instance, I guess way one of showing that vector $A$ has the same direction of $cA,c&gt;0$ would be to show that the line containing $A$ has the same slope as the one containing $cA$ (I'm not sure if this approach is correct though). However, how to proceed in the case of $n$ dimensions? I think I must be missing out something basic here, but I really can't seem to get my head around it. Any help will be highly appreciated. Thanks very much in advance. Allow me to get notifications for new Meta posts, like mods do I'm currently helping out on . One of the most valuable things on new beta sites, especially private/early public betas, is Meta participation - see :  Treat Meta Questions Like Gold It’s not unusual for your meta site to receive significantly less traffic than the main site. The problems begin when you meta starts to get ignored by the community and ignored by the moderators. It is particularly disturbing when users reach out in meta and starts talking about improving the site or asking for help… only to be ignored; No follow-up, only deafening silence  Because of this, Diamond mods get a notification in their inbox for each new question on their Meta, helping ensure that each meta post is addressed. However, wouldn't it be great if Diamond mods weren't the only ones that could do this? The post does say that the community needs to be seeing these questions too. As an involved user, I would really like to be notified of these new questions - it would increase the good I could do the site. How about this?:  Of course, it would be opt-in, not opt-out, and per-site. A feature like this would greatly improve the good that enthusiastic site-builders can do for the site, and would, most likely, increase the overall chances of a site's success. How to find chromatic number of the $n$-dimensional hypercube $Q_n$? Title of 80's movie: future-era timetraveling humans go extinct and rescue/kidnap todays air crash victims What is the title of a 80's (i believe?) movie where humans in the far future are going extinct, and are aquiring new humans by going back in time to todays world, kidnapping air travelers that are known to have died that day, from the flying passenger aircraft just before its crashing.  A key visual:    Inside a hangar, a time portal opened, and a tail of a (flying) passenger aircraft extruded into the hangar where a stairs provided access to the back door of the aircraft.  The kidnappers made sure the surviving passengers remain seated and would forget about the whole ordeal. Kidnapped humans got replaced by lookalike dummies, including things they had on them. They tried hard not to alter the timeline. Novel in which the protagonist fades out is forgotten by, and can't interact with, the rest of the world Trying to identify a novel I read years ago (probably in the '80s) in which the protagonist (late teens, early 20s?) is gradually forgotten by and ceases to be able to interact with the rest of the world (all of which gradually fades to gray).  In the beginning, as he is slowly fading out (not realizing it) he goes down to breakfast with his parents, and his mother has only set out two plates (one for his father and one for herself) — she'd forgotten to set one out for him.   He reminds her.  Foreshadowing, eh?   But gradually everyone forgets about him and ceases to be able to hear or see him, and fade to gray.  Some other things apparently similarly get "forgotten" however — e.g., occasionally he finds colored, not gray, cans of food in grocery stores, which he can interact with and eat — so he survives.  He then meets a girl who has similarly been forgotten by the world, and they spend time together and fall in love.  At the end,       he is gradually starting to fade back into the world (she isn't) and she puts writes a note about them and puts it in his pocket, in the hope it will fade back into the world with him.  It does, and he sees it one day — but has no idea what it is or any memory of her.  Sad. Drawing automata for languages I'm trying to draw two automata for these two languages:   For the first one, I know that the minimum is n = 1, m = 1, but I'm having troubles drawing a NFA for it.  The second one the minimum is n = 2, m = 1, but I still don't know how to start the NFA.  I have this for q2: What does character set and collation mean exactly? I can read the MySQL documentation and it's pretty clear. But, how does one decide which character set to use? On what data does collation have an effect?  I'm asking for an explanation of the two and how to choose them. I want to prove that every real polynomial of odd degree has at least one real root, using  the intermediate value theorem.  Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in \mathbb{R}$ and $n \in \mathbb{N}$.  By the fundamental theorem of algebra I know that $P(x)$ has exactly $2n+1$ complex roots, so  $P(x) = (x+r_1)(x+r_2) . . . (x+r_{2n+1})$ for each $r_i \in \mathbb{C}$  I do not know how to complete this but I do know that, at some point, I probably have to show that each root with imaginary part non zero has to come in conjugate pairs, and since $2n+1$ is odd there is at least $1$ root that is imaginary part $0$ and thus real. What's the difference between a microcontroller and a microprocessor? Correctly typesetting a tilde How do I set the tilde (~) character correctly in LaTeX?  There’s a  on Stack Overflow, with a lot of answers. But dare I say it? They all suck:   \textasciitilde is too high, \texttildelow is too low. $\sim$ can be tweaked to look more or less acceptable: {\raise.17ex\hbox{$\scriptstyle\sim$}} – but even that is a hack and uses the wrong symbol.   The point is – the tilde character is definitely an existing glyph in any font I am tempted to use. How can I just access and typeset that character, please? Without resorting to some other, similar-looking glyph that first needs to be nudged into place using a cunning combination of font size and raised boxes?  There must be a possibility to access that glyph directly from the font definition.  To demonstrate, look at this screenshot which attempts simply to display a C++ destructor of a type T (~T):    Compare this with the symbol we get in the browser: ~ – that’s what I want. Monotonicity in argmin function I have the following equation $$f(a)=\text{argmin}_{t\in\mathbb{R}} \sum_{i=1}^n\frac{1}{2}(1-a)(b_i-t)^2+a| b_i-t |, a\in[0,1]$$  How does one prove that the function is monotone in a? I often see questions edited into this form:     Lorem ipsum dolor sit amet blah blah blah paragraph paragraph original question.            UPDATE: The Lorem disappeared and left this message: "new information".   I find this format unaesthetic (since Stack Exchange posts are supposed to be useful in the long term and update boundaries are artifacts of the creation rather than the content) and feel that new information should, whenever possible, be incorporated into the original text “as if it had always been that way” if practical, and pointing out an update should be done mostly in comments.  Is this an appropriate writing style, and if so under what circumstances? If not, should it be removed or discouraged? How to customize Windows 10 "6 Folders" in My Computer I would like to customize the folders in the Default Folders group of My computer. It is not necessary to have the same folders appear on the left sidebar and in the middle view. How can I do that ? I use the code below to produce handouts of my presentations.  When I open the resulting pdf with any pdf viewer, it is viewed correctly, but when I print it, it is always printed very small, no matter the program I use (Okular, Evince, Acrobat...). Bonus to anyone that can make the pdf be printed correctly by Okular under Ubuntu! Thanks!  MWE here:  \documentclass[handout]{beamer} \usepackage{pgfpages} \pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm] \setbeameroption{show notes on second screen=bottom}  \setbeamercolor{background canvas}{bg=pink}  \mode&lt;presentation&gt; {     \usetheme{Singapore} } \begin{document}   \begin{frame}[plain]     \titlepage   \note{     Notes for the title page } \end{frame}   \section*{Outline} \begin{frame}[plain] \frametitle{Outline}     \tableofcontents  \note{     Notes for the table of contents } \end{frame}   EDIT:   If I place the line "\pgfpagesuselayout{2 on 1}" after the line "\setbeameroption{show notes}" I do not experience the printing problem anymore!! However, the slide preview and notes that appear at the bottom belong to the NEXT SLIDE, instead of to the CURRENT SLIDE!! What is going on with that? Two similar limit problems I have two limit problems which are quite similar, so I've put them both into this post.   $((4^{10}+2^{n})^{\frac{1}{n}})$ $((3n^{2}+n)^{\frac{1}{n}})$   Attempt:  For 1, I'm not sure if this is permissible. I know that $\lim_{n \to \infty}(x^{n}+y^{n})^{\frac{1}{n}} = \max\{x,y\}$. Obviously $4^{10}$ is a constant so it's not quite in the same form, but I believe it is still okay to use this result, and so the limit is $2$.  For 2, I've got $((3n^{2}+n)^{\frac{1}{n}}) = (n(3n+1)^{\frac{1}{n}})$, but then I'm not too sure how to proceed.   I also thought of taking the log and proceeding that way, however in the notes I'm learning from they haven't covered l'hopital yet, so I'm trying to not use it.  Thanks. How do I see that $$f_{n+1}f_{n-1} = f_n^2 + (-1)^n$$ for $n \ge 2$, where $f_1 = 1$, $f_2 = 1$, and $f_{n+2} = f_{n+1} + f_n$ for $n \in \mathbb{N}$? Edelstein Theorem Let $(M,d)$ be a compact metric space and $d(f(x),f(y)) &lt; d(x,y) $ for all $ x\neq y$ Prove that if $f$ is a continuous fuction then there is a unique $x_0 \in M $ such that $f(x_0)=x_0$ I know that since $g(x) = d((f(x),f(y))$ is continuous then a continuous function in a compact space reaches its maximum and minimum. Then by proving that the minimum of $g(x)$ = 0 can I conclude that $x_0$ is a fixed point? find the number of injections/surjections I've been banging my head on this problem for some time now, and could really use help. Bear in mind I'm not very good at this sort of thing and am struggling to get by in class.  Problem:     Given $ \alpha : A \to B       \;\;\;\;\;   |A|=p \;\;\;\;|B|=q $      Find:         Number of Injections: $A \to B$   Number of Surjections: $ A \to B$    Prove that ∃ a bijection $A \to B$       $\Rightarrow$ p = q   find number of bijections: $A \to B$         I apologize for dropping such a big problem on you all, but I just can't get anywhere with this. Help?  EDIT: These answers are really helpful and I believe I've got down the number of Injections, but  we haven't gone through permutations or combinatorics in class so my understanding of it is really hamstrung. I'm unclear on the Stirling theorem as discussed here: , specifically the part where inclusion-exclusion comes into play. I'm not sure how to apply that, since right now I'm letting p = 4 and q = 3, so I don't know where inclusion-exclusion would be used (if at all!) since I don't know how elements would be missing.