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added Category:Water streams using HotCat

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	~~Schur~~'s ~~lemma is a result in Riemannian geometry that says, heuristically, whenever certain curvatures are pointwise constant then they are forced~~ to ~~be globally constant~~. ~~It is essentially a degree of freedom counting argument~~.		It involves expertise and knowledge of various tools and technologies used for creating websites. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. SEO Ultimate - I think this plugin deserves more recognition than it's gotten up till now. Keep reading for some great Word - Press ideas you can start using today. Should you have just about any concerns with regards to exactly where and how you can work with [http://www.l2frog.de/index.php?mod=users&action=view&id=95019 wordpress backup plugin], you'll be able to email us in our web site. You can easily customize the titles of the posts in Word - Press blog in a way that only title comes in the new post link and not the date or category of posts. <br><br>Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. Wordpress have every reason with it which promote wordpress development. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience. <br><br>Photography is an entire activity in itself, and a thorough discovery of it is beyond the opportunity of this content. But if you are not willing to choose cost to the detriment of quality, originality and higher returns, then go for a self-hosted wordpress blog and increase the presence of your business in this new digital age. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. You or your web designer can customize it as per your specific needs. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog. <br><br>The disadvantage is it requires a considerable amount of time to set every thing up. * Robust CRM to control and connect with your subscribers. When we talk about functional suitability, Word - Press proves itself as one of the strongest contestant among its other rivals. If you choose a blog then people will be able to post articles on your site and people will be able to make comments on your posts (unless you turn comments off). Look for experience: When you are searching for a Word - Press developer you should always look at their experience level. <br><br>Instead, you can easily just include it with our bodies integration field in e - Panel. If you operate a website that's been built on HTML then you might have to witness traffic losses because such a site isn't competent enough in grabbing the attention of potential consumers. You can select color of your choice, graphics of your favorite, skins, photos, pages, etc. Change the entire appearance of you blog using themes with one click. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals.

	== ~~Statement~~ of the ~~Lemma ==~~

	~~Suppose <math>(M^n,g)^{}_{}</math> is~~ a ~~[[Riemannian manifold]]~~ and <~~math~~>~~n \geq 3~~<~~/math~~>. ~~Then~~ if

	* the ~~[[sectional curvature]] is pointwise constant~~, ~~that is~~, ~~there exists some function~~ <~~math~~>~~f:M \rightarrow \mathbb{R}~~<~~/math~~> ~~such that~~
	~~:<math>\mathrm{sect}^{}_{}(\Pi_p) = f(p)</math> for all two-dimensional subspaces <math>\Pi_p \subset T_p M</math> and all <math>p \~~in ~~M,</math>~~
	~~:then <math>f</math> is constant~~, and ~~the manifold has constant sectional curvature (also known as~~ a ~~[[space form]] when <math>M</math>~~ is ~~complete); alternatively~~
	* the ~~[[Ricci curvature]] [[endomorphism]] is pointwise a multiple~~ of the ~~identity~~, ~~that is~~, ~~there exists~~ some ~~function <math>f:M \rightarrow \mathbb{R}</math> such that~~
	:~~<math> \mathrm{Ric}^{}_{}~~(~~X_p) = f(p~~) ~~X_p</math>~~ for ~~all <math> X_p \in T_p M</math>~~ and ~~all <math>p \in M~~,<~~/math~~>
	~~:then~~ <~~math>f</math~~> is ~~constant~~, ~~and~~ the ~~manifold is [[Einstein_manifold\|Einstein]]~~.

	~~The requirement that <math>n \geq 3</math> cannot~~ be ~~lifted~~. ~~This result is far from true on two~~-~~dimensional surfaces~~. ~~In two dimensions sectional curvature is always pointwise constant since there is only one two-dimensional subspace~~ <~~math~~>~~\Pi_p \subset T_p M~~<~~/math~~>, ~~namely <math>T_p M</math>~~. ~~Furthermore,~~ in ~~two dimensions~~ the ~~Ricci curvature endomorphism is always a multiple~~ of ~~the identity (scaled by [[Gauss curvature]])~~. ~~On the other hand~~, ~~certainly not all two-dimensional surfaces have constant sectional (or Ricci) curvature~~.


	~~== Reference ==~~
	*S. ~~Kobayashi~~, ~~K. Nomizu. ''[[Foundations of Differential Geometry]]'' (Wiley Classics Library) Volume 1~~, ~~page 202~~.




	~~[[Category:Theorems in Riemannian geometry]]~~
	~~[[Category:Lemmas]]~~

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Schur's lemma is a result in Riemannian geometry that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. It is essentially a degree of freedom counting argument.

== Statement of the Lemma ==

Suppose <math>(M^n,g)^{}_{}</math> is a [[Riemannian manifold]] and <math>n \geq 3</math>. Then if

* the [[sectional curvature]] is pointwise constant, that is, there exists some function <math>f:M \rightarrow \mathbb{R}</math> such that
:<math>\mathrm{sect}^{}_{}(\Pi_p) = f(p)</math> for all two-dimensional subspaces <math>\Pi_p \subset T_p M</math> and all <math>p \in M,</math>
:then <math>f</math> is constant, and the manifold has constant sectional curvature (also known as a [[space form]] when <math>M</math> is complete); alternatively
* the [[Ricci curvature]] [[endomorphism]] is pointwise a multiple of the identity, that is, there exists some function <math>f:M \rightarrow \mathbb{R}</math> such that
:<math> \mathrm{Ric}^{}_{}(X_p) = f(p) X_p</math> for all <math> X_p \in T_p M</math> and all <math>p \in M,</math>
:then <math>f</math> is constant, and the manifold is [[Einstein_manifold|Einstein]].

The requirement that <math>n \geq 3</math> cannot be lifted. This result is far from true on two-dimensional surfaces. In two dimensions sectional curvature is always pointwise constant since there is only one two-dimensional subspace <math>\Pi_p \subset T_p M</math>, namely <math>T_p M</math>. Furthermore, in two dimensions the Ricci curvature endomorphism is always a multiple of the identity (scaled by [[Gauss curvature]]). On the other hand, certainly not all two-dimensional surfaces have constant sectional (or Ricci) curvature.

== Reference ==
*S. Kobayashi, K. Nomizu. ''[[Foundations of Differential Geometry]]'' (Wiley Classics Library) Volume 1, page 202.

[[Category:Theorems in Riemannian geometry]]
[[Category:Lemmas]]

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en>Funandtrvl: added Category:Water streams using HotCat

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