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In [[Riemannian geometry]], the '''fundamental theorem of Riemannian geometry''' states that on any [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]]) there is a unique [[torsion (differential geometry)|torsion-free]] metric [[affine connection|connection]], called the '''[[Levi-Civita connection]]''' of the given metric. Here a '''metric''' (or '''Riemannian''') connection is a connection which preserves the [[metric tensor]]. More precisely:
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<blockquote>'''Fundamental Theorem of Riemannian Geometry.''' Let (''M'', ''g'') be a [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]]). Then there is a unique connection ∇ which satisfies the following conditions:
*for any vector fields ''X'', ''Y'', ''Z'' we have
::<math>\partial_X \langle Y,Z \rangle = \langle \nabla_X Y,Z  \rangle + \langle Y,\nabla_X Z \rangle,</math>
:where <math> \partial_X \langle Y,Z \rangle </math> denotes the derivative of the function <math> \langle Y,Z \rangle </math> along vector field ''X''.
*for any vector fields ''X'', ''Y'',
::<math>\nabla_XY-\nabla_YX=[X,Y],</math>
:where [''X'', ''Y''] denotes the [[Lie bracket of vector fields|Lie bracket]] for [[vector field]]s ''X'', ''Y''. </blockquote>
 
The first condition means that the metric tensor is preserved by [[parallel transport]], while the second condition expresses the fact that the [[torsion (differential geometry)|torsion]] of ∇ is zero.
 
An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the [[metric tensor]] with any given vector-valued 2-form as its torsion.
 
The following technical proof presents a formula for [[Covariant derivative#Coordinate description|Christoffel symbol]]s of the connection in a local coordinate system. For a given metric this set of equations can become rather complicated. There are quicker and simpler methods to obtain the Christoffel symbols for a given metric, e.g. using the [[action (physics)|action]] integral and the associated Euler-Lagrange equations.
 
==Proof==
Let ''m'' be the dimension of ''M'' and, in some local chart, consider the standard coordinate vector fields
 
:<math>{\partial}_i = \frac{\partial}{\partial x^i}, \qquad i=1,\dots,m. </math>
 
Locally, the entry ''g<sub>ij</sub>'' of the metric tensor is then given by 
 
:<math>g_{i j} = \left \langle {\partial}_i, {\partial}_j \right \rangle.</math>
 
To specify the connection it is enough to specify, for all ''i'', ''j'', and ''k'',  
 
:<math>\left \langle \nabla_{\partial_i}\partial_j, \partial_k \right \rangle.</math>
 
We also recall that, locally, a [[Covariant derivative#Coordinate description|connection]] is given by ''m''<sup>3</sup> smooth functions
 
:<math>\left \{ \Gamma^l {}_{ij} \right \},</math>
 
where
 
:<math>\nabla_{\partial_i} \partial_j = \sum_l \Gamma^l_{ij} \partial _l.</math>  
 
The torsion-free property means 
 
:<math>\nabla_{ \partial _i} \partial _j = \nabla_{\partial_j} \partial_i.</math>
 
On the other hand, compatibility with the Riemannian metric implies that
 
:<math> \partial_k g_{ij}  =  \left \langle \nabla_{\partial_k}\partial_i, \partial_j \rangle + \langle \partial_i, \nabla_{\partial_k} \partial_j \right \rangle.</math>
 
For a fixed, ''i'', ''j'', and ''k'', permutation gives 3 equations with 6 unknowns. The torsion free assumption reduces the number of variables to 3. Solving the resulting system of 3 linear equations gives unique solutions
 
:<math>\left \langle \nabla_{ \partial_i }\partial_j, \partial_k \right \rangle  = \tfrac{1}{2} \left ( \partial_i g_{jk}- \partial_k g_{ij} + \partial_j g_{ik} \right ).</math>
 
This is the '''first Christoffel identity'''.
 
Since
 
:<math>\left \langle \nabla_{ \partial_i }\partial_j, \partial_k \right \rangle = \Gamma^l _{ij} g_{lk},</math>
 
where we use Einstein summation convention.  That is, an index repeated [[subscript and superscript]] implies that it is summed over all values. Inverting the metric tensor gives the '''second Christoffel identity''':
 
:<math>\Gamma^l_{ij} = \tfrac{1}{2} \left ( \partial_i g_{jk}- \partial_k g_{ij} + \partial_j g_{ik} \right ) g^{kl}.</math>
 
Once again, with Einstein summation convention.  The resulting unique connection is called the '''Levi-Civita connection'''.
 
==The Koszul formula==
An alternative proof of the Fundamental theorem of Riemannian geometry proceeds by showing that a torsion-free metric connection on a Riemannian manifold is necessarily given by the '''Koszul formula''':
 
:<math>2 g(\nabla_XY, Z) = \partial_X (g(Y,Z)) + \partial_Y (g(X,Z)) - \partial_Z (g(X,Y))+ g([X,Y],Z) - g([X,Z],Y) - g([Y,Z],X).</math>
 
This proves the uniqueness of the Levi-Civita connection. Existence is proven by showing that this expression is tensorial in ''X'' and ''Z'', satisfies the Leibniz rule in ''Y'', and that hence defines a connection. This is a metric connection, because the symmetric part of the formula in ''Y'' and ''Z'' is the first term on the first line; it is torsion-free because the anti-symmetric part of the formula in ''X'' and ''Y'' is the first term on the second line.
 
==See also==
* [[Nash embedding theorem]]
 
{{Fundamental theorems}}
 
{{DEFAULTSORT:Fundamental Theorem Of Riemannian Geometry}}
[[Category:Connection (mathematics)]]
[[Category:Theorems in Riemannian geometry]]
[[Category:Articles containing proofs]]
[[Category:Fundamental theorems|Riemannian geometry]]

Revision as of 21:31, 19 February 2014

Having checked for ovulation, there are some other things that an individual can do to increase the chances of getting pregnant naturally. If a woman wants to have a quick pregnancy, it's always handy to have an inexpensive ovulation test kit. Getting blood to flow can be difficult when a woman is pregnant and by not moving your legs enough, you can get blood clots, which can be deadly. For both women and men, the anatomical and hormonal factors are not the only ones to be monitored and corrected. This really is essential to the majority of us, therefore it is not a thing we must give up on quickly.

Especially, do not take any medications without the advice from your family doctors. Sims are able to have sex with members of the opposite sex as well as members of the same sex. Even though it sounds crazy, the best option would be to give teens some form of birth control and actually allow them to have sex in their house. Woman ovulate about the 14th day so having intercourse during that time is crucial if you want to conceive. They all will certainly reap the benefits of Pregnancy Miracle.

 Dietary Do’s - Dietary Don’ts - Always check with your doctor before you starting any supplements. Conceiving a child maybe easy and simple of all things, yet, some married couples do find it hard to conceive for many reasons, this includes weak or low sperm count. For those considering an alternative medicine option, it's possible to implement forms of relaxing meditation, a healthy diet with proper exercise, adequate sleep and proper posture. Use techniques like meditation, breathing exercises, and yoga. Now, let's talk about Getting Pregnant Plan created by Michelle Adams and Ashley Spencer and how it might help you.

A very old tradition of natural healing using herbal remedies is recognized as the Traditional Chinese Medicines (TCM). Lisa Olson, a certified nutritionist, health consultant and author has not just pumped out yet another "pregnancy program" into an already over-saturated market. When you are in great shape, your body can do what it is designed to do without a problem. The luteal phase and begins the day of ovulation and ends on the day you begin to menstruate. Another trick to help you get pregnant quick is to lie still with your pelvic area lifted by a pillow for at least twenty minutes after intercourse.

It is a shame that every woman cannot born a baby because of infertility problems. But Kim now claims Khloe was never really trying to get pregnant during the past few years, despite the fact that her repeated visits to obstetricians revealed she did have problems conceiving a baby. Couples are affected with the situation of course - they are involved in the condition and some have fallen apart from each other after trying so many methods in getting pregnant tips. To learn about hedgehog facts, hedgehog adoption, caring for hedgehogs, horse facts, horse psychology , horse tips and other information, visit the Interesting Animals website. For women who have a healthy diet, do not need a vitamin supplement, but if you do not feed you properly, then you will be prescribed a multivitamin tablets.

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