A geodesic frame problem Could anyone help me with it? I don`t know how to write by Latex, so I have update a picture of the problem.      thankyou very much!
Existence of a local geodesic frame Let $(M,g)$ be a Riemannian manifold of dimension $n$ with Riemannian connection $\nabla,$ and let $p \in M.$ Show that there exists a neighborhood $U \subset M$ of $p$ and $n$ (smooth) vector fields $E_1,...,E_n \in \chi (U),$ orthonormal at each point of $U,$ s.t. at $p,$ $\nabla_{E_i}E_j(p)=0.$  What I've found so far;  Let $V$ be a normal neighborhood at $p.$ Let $(W,x(x_1,...,x_n))$ be a local coordinate system at $p$ s.t. $x(W) \subset V.$  Let $\{X_i:=\frac{\partial}{\partial x_i}\}$ be the standard basis of $T_pM.$ By Gram-Schmidt, there exists an orthonormal basis $\{E_i\}$ of $T_pM.$ Consider the orthonormal frame $E_i(t),$ by parallel transporting $\{E_i(0)=E_i\}$ along a  a geodesic $\gamma :I \to M$ starting at $\gamma(0)=p$ and ending in $\gamma(1)=q,$ where $q \in V.$ This gives us an orthonormal frame at each point of $V.$ To satisfy the second condition, let $E_i(t)=\sum_{l}a_{li}X_l$ and  $E_j(t)=\sum_{s}b_{sj}(t)X_s.$ Then   $$\nabla_{E_i}E_j=\sum_{k}(\sum_{l,s} a_{li}(t)b_{sj}(t)\Gamma^k_{ls}+E_i(b_{kj}))X_k,$$  where $\Gamma^k_{is}$ are Christoffel symbols.   To have $\nabla_{E_i}E_j(p)=0,$ we must have $\sum_{l,s}a_{li}(0)b_{sj}(0)\Gamma^k_{ls}+E_i(b_{kj})(p)=0$ for each $k.$      How can I conclude the argument?
After reading an article about Ubuntu's netflix support I decided to try running netflix through chromium but I got this error code: M7063-1913 I assumed that it was a problem with html5 but in the playback preferences it doesn't give me the option to use it. I am a precision modeler, which means I like to be exact instead of "eyeballing" things when I model. I am very happy to see the option to Edge Slide with Clamping Off in v2.75a, but I don't seem to be able to snap the edge slide destination to a vertex point elsewhere on the model. Is this possible and I am just missing something? Creating Layer in GeoServer using REST As part of a possible solution to   I want create a layer in GeoServer via REST that supplies a SQL statement along with a userid. I think i would  and publish it. I need to do this all in REST. Does anyone have any code examples? Motivating question:    I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process:  $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \ldots $$  "Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions" $\subset$ $\ldots$  and that at each step you're given a multiplication table that tell how the elements interact. As you move up the ladder, certain "nice" properties are lost: ordering,  commutativity, associativity, multiplicative normedness, etc... Given the multiplication table, you can show that these properties don't hold.   Eric Naslund noted that "the first 4 are very special as they are the unique 4 normed division algebras over ℝ", no surprise then that these $2^n$-ions have found quite a bit of use. I'm interested in the sequence itself however, irrespective of how useful a $2^{256}$-ion might be (ducenti-quinquaginta-sex-ion?).   I feel like something deeper is going on here though that I don't understand. Why are these particular properties lost at each step? Is it possible to quantify the process such that, at the $2^n$-ion you can say something about the symmetry of the multiplication table*?  * I'm making an ansatz that there is a connection between the symmetry of the multiplication table and these "nice" properties. What is the process for a moderator to request reinstatement or appeal a removal? Number of triangles formed by $m$ lines Problem:    How many triangles do $m$ lines form if   a) Every two lines intersect and no three lines intersect at one point.  b) There are $n$ lines among $m$ lines that are parallel to each other. No other line is parallel to these $n$ lines, and no other two lines are parallel to each other. Again no three lines intersect at one point.  Thank you. I'd like to install a  client for Xubuntu (12.04).  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Since $1\in I$ and $I\subseteq R$ is an ideal, then let $a=1$ and $r=1$ so $1\cdot 1\in R\,?$ What's the meaning of a dot before a command in shell? While following android eclipse debug tutorial, I encounter following commands.  cd /path/to/android/root  . build/envsetup.sh  lunch 1     make        emulator   My problem is what the dot before build/envsetup.sh means? Is it possible to restore a MySql database only using files from Data directory? I'm trying to load a field value (machine name: field_theme_image) from a content of node value 3, in "page--front.tpl.php".  &lt;?php   $node=node_load(3);   print $node-&gt;field_theme_image[$node-&gt;language][0]['uri']; ?&gt;   Using the above code I'm not getting full URL value. Please see the image below. I can see the path value in Devel themer. Is it possible to dual boot two OS's at the same time? reCaptcha v1 is shutting down on March 31, but SEDE still uses it According to this , I thought the new recaptcha API was used network-wide. Though today, when going to  (which for whatever reasons never recognized me automatically), I got this message instead of a usable captcha image.   V1 Unsupported Please direct siteowner to   (Doing a second try doesn't show this message anymore, and the recaptcha v1 works as intended.)  According to , the reCAPTCHA v1 API will stop working on March 31, 2018:  What happens to reCAPTCHA v1? Any calls to the v1 API will not work after March 31, 2018. Starting in November 2017, a percentage of reCAPTCHA v1 traffic will begin to show a notice informing users that the old API will soon be retired. Most websites around the world have already switched to reCAPTCHA v2, where humans pass effortlessly. With our advanced risk analysis engine, reCAPTCHA v2 can effectively separate humans from bots and always stay ahead of the attackers. If your site still uses reCAPTCHA v1, please upgrade to reCAPTCHA v2 immediately. We no longer support reCAPTCHA v1. Any calls to the v1 API will not work after March 31, 2018. Please register a new key and upgrade to v2.  I'm really hoping this will not take  to fix, because we've only got three and a half weeks left! Why many abstract nouns get an indefinite article while it's taught that abstract nouns shouldn't get it? Will my Chinese visa still be valid if I renew my passport? Will my Chinese visa still be valid if I renew my passport? If so how does that work?  I have heard that you can travel with your new passport and just show the old passport with the valid visa when entering China. Has anyone done that? Recently?   I have also been told that you have to get a new visa, which seems somewhat logical but frustrating, especially when my Chinese visa is only a few months old.  My Chinese visa was issued in late Nov 2014 and I heard the rules changed between the US and China. My visa says it's good for 10 years and my US passport expires in 5. Skype window problem in Ubuntu 14.04 On the canonical isomorphism between $V$ and $V^{**}$ When is $-3$ a quadratic residue mod $p$?