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| In [[Riemannian geometry]], the '''Levi-Civita connection''' is a specific [[connection (mathematics)|connection]] on the tangent bundle of a [[manifold]]. More specifically, it is the [[Torsion (differential geometry)|torsion]]-free [[metric connection]], i.e., the torsion-free [[connection (mathematics)|connection]] on the [[tangent bundle]] (an [[affine connection]]) preserving a given ([[pseudo-Riemannian manifold|pseudo-]])[[Riemannian metric]].
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| The [[fundamental theorem of Riemannian geometry]] states that there is a unique connection which satisfies these properties.
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| In the theory of [[Riemannian manifold|Riemannian]] and [[pseudo-Riemannian manifold]]s the term [[covariant derivative]] is often used for the Levi-Civita connection. The components of this connection with respect to a system of local coordinates are called [[Christoffel symbols]]. | |
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| The Levi-Civita connection is named after [[Tullio Levi-Civita]], although originally "discovered" by [[Elwin Bruno Christoffel]]. Levi-Civita,<ref>See Levi-Civita (1917)</ref> along with [[Gregorio Ricci-Curbastro]], used Christoffel's symbols<ref>See Christoffel (1869)</ref> to define the notion of [[parallel transport]] and explore the relationship of parallel transport with the [[Riemann curvature tensor|curvature]], thus developing the modern notion of [[holonomy]].<ref>See Spivak (1999) Volume II, page 238</ref>
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| The Levi-Civita notions of intrinsic derivative and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific embedding
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| :<math>M^n \subset \mathbf{R}^{\frac{n(n+1)}{2}},</math>
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| since the definition of the Christoffel symbols make sense in any Riemannian manifold. In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis. It was not until 1917 that Levi-Civita interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine space.
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| ==Formal definition==
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| Let ''(M,g)'' be a [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]]). Then an [[affine connection]] ∇ is called a Levi-Civita connection if
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| # ''it preserves the metric'', i.e., {{nowrap|1=∇''g'' = 0}}.
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| # ''it is [[torsion of connection|torsion]]-free'', i.e., for any vector fields ''X'' and ''Y'' we have {{nowrap|1=∇<sub>''X''</sub>''Y'' − ∇<sub>''Y''</sub>''X'' = [''X'',''Y'']}}, where [''X'',''Y''] is the [[Lie bracket of vector fields|Lie bracket]] of the [[vector field]]s ''X'' and ''Y''.
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| Condition 1 above is sometimes referred to as [[compatibility with the metric]], and condition 2 is sometimes called symmetry, cf. DoCarmo's text.
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| Assuming a Levi-Civita connection exists it is uniquely determined. Using conditions 1 and the symmetry of the metric tensor ''g'' we find:
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| :<math> X (g(Y,Z)) + Y (g(Z,X)) - Z (g(Y,X)) = g(\nabla_X Y + \nabla_Y X, Z) + g(\nabla_X Z - \nabla_Z X, Y) + g(\nabla_Y Z - \nabla_Z Y, X). </math>
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| By condition 2 the right hand side is equal to
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| :<math> 2g(\nabla_X Y, Z) - g([X,Y], Z) + g([X,Z],Y) + g([Y,Z],X) </math>
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| so we find
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| :<math> g(\nabla_X Y, Z) = \frac{1}{2} \{ X (g(Y,Z)) + Y (g(Z,X)) - Z (g(X,Y)) + g([X,Y],Z) - g([Y,Z], X) - g([X,Z], Y) \}. </math>
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| Since ''Z'' is arbitrary, this uniquely determines ∇<sub>''X''</sub>''Y''. Conversely, using the last line as a definition one shows that the expression so defined is a connection compatible with the metric, i.e. is a Levi-Civita connection.
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| ==Christoffel symbols==
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| Let ∇ be the connection of the Riemannian metric. Choose local coordinates <math> x^1 \ldots x^n</math> and let <math> \Gamma^l{}_{jk} </math> be the [[Christoffel symbols]] with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry
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| :<math> \Gamma^l{}_{jk} = \Gamma^l{}_{kj}. </math>
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| The definition of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as
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| :<math> \Gamma^l{}_{jk} = \tfrac{1}{2}\sum_r g^{lr} \left \{\partial _k g_{rj} + \partial _j g_{rk} - \partial _r g_{jk} \right \} </math>
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| where as usual <math>g^{ij}</math> are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix <math>(g_{kl})</math>.
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| ==Derivative along curve==
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| The Levi-Civita connection (like any affine connection) also defines a derivative along [[curve]]s, sometimes denoted by ''D''.
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| Given a smooth curve γ on ''(M,g)'' and a [[vector field]] ''V'' along γ its derivative is defined by
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| :<math>D_tV=\nabla_{\dot\gamma(t)}V.</math>
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| (Formally ''D'' is the [[pullback (differential geometry)|pullback connection]] on the [[pullback bundle]] γ*''TM''.)
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| In particular, <math>\dot{\gamma}(t)</math> is a vector field along the curve γ itself. If <math>\nabla_{\dot\gamma(t)}\dot\gamma(t)</math> vanishes, the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those [[geodesics]] of the [[Metric tensor|metric]] that are parametrised proportionally to their arc length.
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| ==Parallel transport==
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| In general, [[parallel transport]] along a curve with respect to a connection defines isomorphisms between the tangent spaces at the points of the curve. If the connection is a Levi-Civita connection, then these isomorphisms are [[Orthogonal group|orthogonal]] – that is, they preserve the inner products on the various tangent spaces.
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| ==Example: The unit sphere in R<sup>3</sup>==
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| Let <math>\langle \cdot,\cdot \rangle</math> be the usual [[scalar product]] on '''R'''<sup>3</sup>. Let '''S'''<sup>2</sup> be the [[unit sphere]] in '''R'''<sup>3</sup>. The tangent space to '''S'''<sup>2</sup> at a point ''m'' is naturally identified with the vector sub-space of '''R'''<sup>3</sup> consisting of all vectors orthogonal to ''m''. It follows that a vector field ''Y'' on '''S'''<sup>2</sup> can be seen as a map ''Y'': '''S'''<sup>2</sup> → '''R'''<sup>3</sup>, which satisfies
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| :<math>\langle Y(m), m\rangle = 0, \qquad \forall m\in \mathbf{S}^2.</math>
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| Denote by ''dY'' the differential of such a map. Then we have:
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| <blockquote>'''Lemma:''' The formula
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| :<math>\left(\nabla_X Y\right)(m) = d_mY(X) + \langle X(m),Y(m)\rangle m</math>
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| defines an affine connection on '''S'''<sup>2</sup> with vanishing torsion.</blockquote>
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| '''Proof:''' It is straightforward to prove that ∇ satisfies the Leibniz identity and is ''C''<sup>∞</sup>('''S'''<sup>2</sup>) linear in the first variable. It is also a straightforward computation to show that this connection is torsion free. So all that needs to be proved here is that the formula above does indeed define a vector field. That is, we need to prove that for all ''m'' in '''S'''<sup>2</sup>
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| :<math>\langle\left(\nabla_X Y\right)(m),m\rangle = 0\qquad (1).</math>
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| Consider the map
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| :<math>\begin{cases}
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| f: \mathbf{S}^2 \to \mathbf{R} \\
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| m \mapsto \langle Y(m), m\rangle.
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| \end{cases}</math>
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| The map ''f'' is constant, hence its differential vanishes. In particular
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| :<math>d_mf(X) = \langle d_m Y(X),m\rangle + \langle Y(m), X(m)\rangle = 0.</math>
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| The equation (1) above follows.<math>\Box</math> | |
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| In fact, this connection is the Levi-Civita connection for the metric on '''S'''<sup>2</sup> inherited from '''R'''<sup>3</sup>. Indeed, one can check that this connection preserves the metric.
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| ==See also==
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| *[[Affine connection]]
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| *[[Weitzenböck connection]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| ===Primary historical references===
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| * {{citation | first = Elwin Bruno | last = Christoffel |title= Über die Transformation der homogenen Differentialausdrücke zweiten Grades| journal = J. für die Reine und Angew. Math.| volume = 70 | year = 1869 | pages = 46–70}}
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| * {{citation | first = Tullio | last = Levi-Civita |title= Nozione di parallelismo in una varietà qualunque e consequente specificazione geometrica della curvatura Riemanniana| journal = Rend. Circ. Mat. Palermo| volume = 42 | year = 1917 | pages = 73–205}}
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| ===Secondary references===
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| * {{cite book|first=William M.|last=Boothby|title=An introduction to differentiable manifolds and Riemannian geometry |publisher=Academic Press|year=1986|isbn=0-12-116052-1}}
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| * {{cite book|author=Kobayashi, S., and Nomizu, K.|title=Foundations of differential geometry|publisher=John Wiley & Sons|year=1963|isbn=0-470-49647-9}} See Volume I pag. 158
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| * {{cite book|first=Michael|last=Spivak|title=A Comprehensive introduction to differential geometry (Volume II)|publisher=Publish or Perish Press|year=1999|isbn=0-914098-71-3}}
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| ==External links==
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| * {{springer|title=Levi-Civita connection|id=p/l058230}}
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| * [http://mathworld.wolfram.com/Levi-CivitaConnection.html MathWorld: Levi-Civita Connection]
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| * [http://planetmath.org/encyclopedia/LeviCivitaConnection.html PlanetMath: Levi-Civita Connection]
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| * [http://www.map.mpim-bonn.mpg.de/Levi-Civita_connection Levi-Civita connection] at the Manifold Atlas
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| {{Tensors}}
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| {{DEFAULTSORT:Levi-Civita Connection}}
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| [[Category:Riemannian geometry]]
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| [[Category:Connection (mathematics)]]
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The task of home coffee roasting is decades old. It has gone on for centuries whenever it was started by roasting green coffee beans over an open fire inside pans. It eventually went to roasting over coals inside cast iron skillets, or rotating iron drums over a fire. Up until regarding World War I, it was prevalent to roast coffee at house. In the 20th century, commercial roasting plus grinding moved in to the scene and it became less prevalent for people to roast their own coffee.
It can be. However, due to the popularity of the green coffee bean extract extract, those firms creating top quality extracts inside the U.S., containing at least 50% chlorogenic acid, can't discover the supply to meet the consumer demand! So the statement watched on countless sites that they will run out of product, is not only a sales tactic. Running from supply arises quite frequently.
The thought of eating green beans extract to begin losing fat may appear really modern nonetheless it has assisted many people get back to their excellent weight and more. Reducing weight is never a simple thing to shoot for. Apart from the undeniable truth which it'll include a wonderful deal of control plus function, 1 is never really sure when the several products being presented on the market will succeed or-not. The problem now is, could this extract be an ideal way to lose excess fat? Can it be safe? How swiftly could you certainly see results? Keep reading and discover.
Instead, choose fruits and veggies for your carbohydrates. Another superior way to consume is any lean meats. Meats like chicken and fish are greater options than fatty red meats. Whenever it comes to refreshments, you don't wish To drink the calories. You'll find considerable amounts of sugar and calories in soda plus juices. Create an attempt to drink lower calorie refreshments or low fat milk in between or with meals. A terrific technique to start leading a healthier lifestyle is to include the assistance of all the folks in a family.
In their scientific study trial a group of 16 overweight persons were given GCBE every day. All participants consumed a diet of 2400 calories a day. Although this amount of calories per day is above the recommended average, none of the participants did much exercise plus, amazingly, they nevertheless lost fat. Over a period of 12 weeks the average weight reduction was 17 pounds. This is a 10.5 % decrease in overall body weight along with a 16 % loss inside body fat.
Medium Roast Coffee: The medium roast coffee covers the full scheme of coffees that fall somewhere amidst a light along with a dark roast. It is entirely up to the roaster and the area of the coffee bean, it can-have a medium to full body taste and a smooth whichever or somewhat acidic aftertaste. Its goal is to give the greatest of both the light and the dark roast. It desires to capture the flavor of the green bean.
Gourmet coffee roasters are available for purchase with the click of the mouse. You can purchase these roasters online plus inside many department and specialty stores. What is beautiful about having a roaster is that you are able to roast only the beans you're going to use; therefore, you are drinking the most delicious and freshest coffee possible.
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