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| | Or maybe already know that protective clothing, including riding hats, are needed your horseback riding safety. Although equestrians and others love horses - they are horses beautiful, often friendly and intelligent, and fun to ride - the growing system become dangerous when startles or scared.<br><br><br><br>Answer: a colossal yes. Dog prints an error in judgment can be returned and should be established companies. Remember you are specific changes that many avoid such mistakes however. However, the company re-shoulder the shipping cost and subject of is fees.<br><br>Kid's clothing come in assorted sizes. To means to opt for the appropriate size, you in order to be consider the age, weight, and height of they. There are instances wherein the little child doesn't really follow the standard size because of their age. Nowadays wherein children can often be bigger since age, leads to also capacity to measure the size and have the appropriate clothing for their own needs. Read on for some tips and recommendations.<br><br>Getting dimension right, balanced by the requirements of your hair and equal in shape to your profile takes experimentation. It is time well spent to learn which sizes and shapes are most appropriate for people.<br><br>Ms. Linda MacMahon also talks about health care reform by attacking what she calls the main culprit to high costs and premiums, malpractice. She believes any time we reform malpractice, thats liable to bring on unnecessary testing, medicines, and procedures due to lawsuit liability, we could cut defensive medicine spending by $191 billion, $42 billion in increased output, and $9 billion in tort values. These numbers stripped away from the linda2010 website come from the Pacific Research Institute and have been shocking if you. I always knew that malpractice played a big part of costs in health proper care. Some doctors experienced to quit their practices because of it, nonetheless had not a clue exactly exactly how much it was costing us, the people counting on care.<br><br>Your child may n't need to curl up. They may not want to huged unless built playing all-around. This is unusual in children and might be confirmed if that is the case.<br><br>Continue testing what works for your online shop and improving the factors that do affect conversions and your bottomline. This can be the only method to get it right and continue growing your store.<br><br>When you loved this informative article as well as you would like to acquire more details relating to [http://usmerch.co.uk/accessories/ gas monkey garage] generously go to the site. |
| In [[Riemannian geometry]], the '''scalar curvature''' (or the '''Ricci scalar''') is the simplest [[curvature]] invariant of a [[Riemannian manifold]]. To each point on a Riemannian manifold, it assigns a single [[real number]] determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the [[volume]] of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the [[Gaussian curvature]], and completely characterizes the curvature of a surface. In more than two dimensions, however, the [[curvature of Riemannian manifolds]] involves more than one functionally independent quantity.
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| In [[general relativity]], the scalar curvature is the [[Lagrangian]] density for the [[Einstein–Hilbert action]]. The [[Euler–Lagrange equations]] for this Lagrangian under variations in the metric constitute the vacuum [[Einstein field equations]], and the stationary metrics are known as [[Einstein manifold|Einstein metrics]]. The scalar curvature is defined as the trace of the [[Ricci tensor]], and it can be characterized as a multiple of the average of the [[sectional curvature]]s at a point. Unlike the Ricci tensor and sectional curvature, however, global results involving only the scalar curvature are extremely subtle and difficult. One of the few is the [[positive mass theorem]] of [[Richard Schoen]], [[Shing-Tung Yau]] and [[Edward Witten]]. Another is the [[Yamabe problem]], which seeks extremal metrics in a given [[conformal class]] for which the scalar curvature is constant.
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| ==Definition==
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| The scalar curvature is usually denoted by ''S'' (other notations are ''Sc'', ''R''). It is defined as the [[Trace (linear algebra)|trace]] of the [[Ricci curvature]] tensor with respect to the [[metric tensor|metric]]:
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| :<math>S = \mbox{tr}_g\,\operatorname{Ric}.</math>
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| The trace depends on the metric since the Ricci tensor is a (0,2)-valent tensor; one must first [[raising and lowering indices|raise an index]] to obtain a (1,1)-valent tensor in order to take the trace. In terms of [[local coordinates]] one can write
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| :<math>S = g^{ij}R_{ij} = R^j_j</math>
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| where ''R''<sub>''ij''</sub> are the components of the Ricci tensor in the coordinate basis:
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| :<math>\operatorname{Ric} = R_{ij}\,dx^i\otimes dx^j.</math>
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| Given a coordinate system and a metric tensor, scalar curvature can be expressed as follows
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| :<math>S = g^{ab} (\Gamma^c_{ab,c} - \Gamma^c_{ac,b} + \Gamma^d_{ab}\Gamma^c_{cd} - \Gamma^d_{ac} \Gamma^c_{bd})
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| =
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| 2g^{ab} (\Gamma^c_{a[b,c]} + \Gamma^d_{a[b}\Gamma^c_{c]d})
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| </math> | |
| where <math>\Gamma^a_{bc}</math> are the [[Christoffel symbols]] of the metric.
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| Unlike the [[Riemann curvature tensor]] or the [[Ricci tensor]], which both can be naturally defined for any [[affine connection]], the scalar curvature requires a metric of some kind. The metric can be [[pseudo-Riemannian]] instead of Riemannian. Indeed, such a generalization is vital to relativity theory. More generally, the Ricci tensor can be defined in broader class of [[metric geometry|metric geometries]] (by means of the direct geometric interpretation, below) that includes [[Finsler geometry]].
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| ==Direct geometric interpretation==
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| When the scalar curvature is positive at a point,
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| the volume of a small ball about the point has smaller volume than
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| a ball of the same radius in Euclidean space. On the other hand,
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| when the scalar curvature is negative at a point, the volume of a small ball is instead larger than it would be in Euclidean space.
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| This can be made more quantitative, in order to characterize the precise value of the scalar curvature ''S'' at a point ''p'' of a Riemannian n-manifold <math>(M,g)</math>.
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| Namely, the ratio of the ''n''-dimensional volume of a ball of radius ε in the manifold to that of a corresponding ball in
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| Euclidean space is given, for small ε, by
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| : <math> \frac{\operatorname{Vol} (B_\varepsilon(p) \subset M)}{\operatorname{Vol}
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| (B_\varepsilon(0)\subset {\mathbb R}^n)}=
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| 1- \frac{S}{6(n+2)}\varepsilon^2 + O(\varepsilon^4).</math>
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| Thus, the second derivative of this ratio, evaluated at radius ε = 0, is exactly minus the scalar curvature divided by 3(''n'' + 2).
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| Boundaries of these balls are (n-1) dimensional spheres with radii <math>\epsilon</math>; their hypersurface measures ("areas") satisfy the following equation:
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| : <math> \frac{\operatorname{Area} (\partial B_\varepsilon(p) \subset M)}{\operatorname{Area}
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| (\partial B_\varepsilon(0)\subset {\mathbb R}^n)}=
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| 1- \frac{S}{6n}\varepsilon^2 + O(\varepsilon^4).</math>
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| ==Special cases==
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| ===Surfaces===
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| In two dimensions, scalar curvature is exactly twice the [[Gaussian curvature]]. For an embedded surface in [[Euclidean space]], this means that
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| :<math>S = \frac{2}{\rho_1\rho_2}\,</math>
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| where <math>\rho_1,\,\rho_2</math> are [[Principal curvature|principal radii]] of the surface. For example, scalar curvature of a sphere with radius r is equal to 2/''r''<sup>2</sup>.
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| The 2-dimensional [[Riemann tensor]] has only one independent component and it can be easily expressed
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| in terms of the scalar curvature and metric area form. In any coordinate system, one thus has: | |
| :<math>2R_{1212} \,= S \det (g_{ij}) = S[g_{11}g_{22}-(g_{12})^2].</math>
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| ===Space forms===
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| A [[space form]] is by definition a Riemannian manifold with constant sectional curvature. Space forms are locally isometric to one of the following types:
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| * [[Euclidean space]]: The Riemann tensor of an ''n''-dimensional Euclidean space vanishes identically, so the scalar curvature does as well.
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| * [[N-sphere|''n''-spheres]]: The sectional curvature of an ''n''-sphere of radius ''r'' is ''K'' = 1/''r''<sup>2</sup>. Hence the scalar curvature is ''S'' = ''n''(''n''−1)/''r''<sup>2</sup>.
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| * [[Hyperbolic space]]s: By the [[hyperboloid model]], an ''n'' dimensional hyperbolic space can be identified with the subset of (''n''+1)-dimensional [[Minkowski space]]
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| ::<math>x_0^2-x_1^2-\cdots-x_n^2 = r^2,\quad x_0>0.</math>
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| :The parameter ''r'' is a geometrical invariant of the hyperbolic space, and the sectional curvature is ''K'' = −1/''r''<sup>2</sup>. The scalar curvature is thus ''S'' = −''n''(''n''−1)/''r''<sup>2</sup>.
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| ==Traditional notation==
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| Among those who use index notation for tensors, it is common to use the letter ''R'' to represent three different things:
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| #the [[Riemann curvature tensor]]: <math>R_{ijk}^l</math> or <math>R_{abcd}</math>
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| #the [[Ricci tensor]]: <math>R_{ij}</math>
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| #the scalar curvature: ''R''
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| These three are then distinguished from each other by their number of indices: the Riemann tensor has four indices, the Ricci tensor has two indices, and the Ricci scalar has zero indices. Those not using an index notation usually reserve ''R'' for the full Riemann curvature tensor.
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| ==See also==
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| * [[Basic introduction to the mathematics of curved spacetime]]
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| * [[Yamabe invariant]]
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| * [[Kretschmann scalar]]
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| {{Curvature}}
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| {{DEFAULTSORT:Scalar Curvature}}
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| [[Category:Curvature (mathematics)]]
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| [[de:Riemannscher Krümmungstensor#Krümmungsskalar]]
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Or maybe already know that protective clothing, including riding hats, are needed your horseback riding safety. Although equestrians and others love horses - they are horses beautiful, often friendly and intelligent, and fun to ride - the growing system become dangerous when startles or scared.
Answer: a colossal yes. Dog prints an error in judgment can be returned and should be established companies. Remember you are specific changes that many avoid such mistakes however. However, the company re-shoulder the shipping cost and subject of is fees.
Kid's clothing come in assorted sizes. To means to opt for the appropriate size, you in order to be consider the age, weight, and height of they. There are instances wherein the little child doesn't really follow the standard size because of their age. Nowadays wherein children can often be bigger since age, leads to also capacity to measure the size and have the appropriate clothing for their own needs. Read on for some tips and recommendations.
Getting dimension right, balanced by the requirements of your hair and equal in shape to your profile takes experimentation. It is time well spent to learn which sizes and shapes are most appropriate for people.
Ms. Linda MacMahon also talks about health care reform by attacking what she calls the main culprit to high costs and premiums, malpractice. She believes any time we reform malpractice, thats liable to bring on unnecessary testing, medicines, and procedures due to lawsuit liability, we could cut defensive medicine spending by $191 billion, $42 billion in increased output, and $9 billion in tort values. These numbers stripped away from the linda2010 website come from the Pacific Research Institute and have been shocking if you. I always knew that malpractice played a big part of costs in health proper care. Some doctors experienced to quit their practices because of it, nonetheless had not a clue exactly exactly how much it was costing us, the people counting on care.
Your child may n't need to curl up. They may not want to huged unless built playing all-around. This is unusual in children and might be confirmed if that is the case.
Continue testing what works for your online shop and improving the factors that do affect conversions and your bottomline. This can be the only method to get it right and continue growing your store.
When you loved this informative article as well as you would like to acquire more details relating to gas monkey garage generously go to the site.