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| In [[mathematics]], specifically [[differential geometry]], the infinitesimal geometry of [[Riemannian manifold]]s with dimension at least 3 is ''too complicated'' to be described by a single number at a given point. [[Riemann]] introduced an abstract and rigorous way to define it, now known as the curvature [[tensor]]. Similar notions have found applications everywhere in differential geometry.
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| For a more elementary discussion see the article on [[curvature]] which discusses the curvature of curves and surfaces in 2 and 3 dimensions, as well as [[Differential geometry of surfaces]].
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| The curvature of a [[pseudo-Riemannian manifold]] can be expressed in the same way with only slight modifications.
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| == Ways to express the curvature of a Riemannian manifold ==
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| ===The Riemann curvature tensor===
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| {{Main|Riemann curvature tensor}}
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| The curvature of a Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a [[Levi-Civita connection]] (or [[covariant differentiation]]) <math>\nabla</math> and [[Lie derivative|Lie bracket]] <math>[\cdot,\cdot]</math> by the following formula:
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| :<math>R(u,v)w=\nabla_u\nabla_v w - \nabla_v \nabla_u w -\nabla_{[u,v]} w .</math>
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| Here <math>R(u,v)</math> is a linear transformation of the tangent space of the manifold; it is linear in each argument.
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| If <math>u=\partial/\partial x_i</math> and <math>v=\partial/\partial x_j</math> are coordinate vector fields then <math>[u,v]=0</math> and therefore the formula simplifies to
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| :<math>R(u,v)w=\nabla_u\nabla_v w - \nabla_v \nabla_u w </math>
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| i.e. the curvature tensor measures ''noncommutativity of the covariant derivative''.
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| The linear transformation <math>w\mapsto R(u,v)w</math> is also called the '''curvature transformation''' or '''endomorphism'''.
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| '''NB.''' There are a few books where the curvature tensor is defined with opposite sign.
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| ====Symmetries and identities====
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| The curvature tensor has the following symmetries:
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| :<math>R(u,v)=-R(v,u)^{}_{}</math>
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| :<math>\langle R(u,v)w,z \rangle=-\langle R(u,v)z,w \rangle^{}_{}</math>
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| :<math>R(u,v)w+R(v,w)u+R(w,u)v=0 ^{}_{}</math>
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| The last identity was discovered by [[Gregorio Ricci-Curbastro|Ricci]], but is often called the ''first Bianchi identity'', just because it looks similar to the Bianchi identity below. The first two should be addressed as ''antisymmetry'' and ''Lie algebra property'' resp., since the second means, that the R(u, v) for all u, v are elements of the pseudo-orthogonal Lie algebra. All three together should be named ''pseudo-orthogonal curvature structure''. They give rise to a ''tensor'' only by identifications with objects of the tensor algebra - but likewise there are identifications with concepts in the Clifford-algebra. Let us note, that these three axioms of a curvature structure give rise to a well-developed structure theory, formulated in terms of projectors (a Weyl projector, giving rise to ''Weyl curvature'' and an Einstein projector, needed for the setup of the Einsteinian gravitational equations). This structure theory is compatible with the action of the pseudo-orthogonal groups plus dilatations. It has strong ties with the theory of Lie groups and algebras, Lie triples and Jordan algebras. See the references given in the discussion.
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| The three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one could find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has <math>n^2(n^2-1)/12</math> independent components.
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| Yet another useful identity follows from these three:
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| :<math>\langle R(u,v)w,z \rangle=\langle R(w,z)u,v \rangle^{}_{}</math>
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| The '''Bianchi identity''' (often the '''second Bianchi identity''')
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| involves the covariant derivatives:
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| :<math>\nabla_uR(v,w)+\nabla_vR(w,u)+\nabla_w R(u,v)=0</math>
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| ===Sectional curvature===
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| {{Main|Sectional curvature}}
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| Sectional curvature is a further, equivalent but more geometrical, description of the curvature of Riemannian manifolds. It is a function <math>K(\sigma)</math> which depends on a ''section'' <math>\sigma</math> (i.e. a 2-plane in the tangent spaces). It is the [[curvature|Gauss curvature]] of the <math>\sigma </math>-''section'' at ''p''; here <math>\sigma </math>-''section'' is a locally-defined piece of surface which has the plane <math>\sigma </math> as a tangent plane at ''p'', obtained from geodesics which start at ''p'' in the directions of the image of <math>\sigma </math> under the [[exponential map]] at ''p''.
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| If <math>v,u</math> are two linearly independent vectors in <math>\sigma</math> then
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| :<math>K(\sigma)= K(u,v)/|u\wedge v|^2\text{ where }K(u,v)=\langle R(u,v)v,u \rangle</math>
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| The following formula indicates that sectional curvature describes the curvature tensor completely:
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| :<math>6\langle R(u,v)w,z \rangle =^{}_{}</math>
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| :<math>[K(u+z,v+w)-K(u+z,v)-K(u+z,w)-K(u,v+w)-K(z,v+w)-K(v+z,u)+K(u,w)+K(v,z)]-^{}_{}</math>
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| :<math>[K(u+w,v+z)-K(u+w,v)-K(u+w,z)-K(u,v+z)-K(w,v+z)-K(u+w,v)+K(v,w)+K(u,z)].^{}_{} </math>
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| Or in a simpler formula:
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| <math>\langle R(u,v)w,z\rangle=\frac 16 \left.\frac{\partial^2}{\partial s\partial t}
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| \left(K(u+sz,v+tw)-K(u+sw,v+tz)\right)\right|_{(s,t)=(0,0)}</math>
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| ===Curvature form===
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| {{Main|Curvature form}}
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| The [[connection form]] gives an alternative way to describe curvature. It is used more for general [[vector bundle]]s, and for [[principal bundle]]s, but it works just as well for the tangent bundle with the [[Levi-Civita connection]]. The curvature of ''n''-dimensional Riemannian manifold is given by an [[antisymmetric matrix|antisymmetric]] ''n''×''n'' matrix <math>\Omega^{}_{}=\Omega^i_{\ j}</math> of [[2-form]]s (or equivalently a 2-form with values in <math>so(n)</math>, the [[Lie algebra]] of the [[orthogonal group]] <math>O(n)</math>, which is the [[structure group]] of the tangent bundle of a Riemannian manifold).
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| Let <math>e_i</math> be a local section of orthonormal bases. Then one can define the connection form, an antisymmetric matrix of 1-forms <math>\omega=\omega^i_{\ j}</math> which satisfy from the following identity
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| :<math>\omega^k_{\ j}(e_i)=\langle \nabla_{e_i}e_j,e_k\rangle</math>
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| Then the curvature form <math>\Omega=\Omega^i_{\ j}</math> is defined by
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| :<math>\Omega=d\omega +\omega\wedge\omega</math>
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| The following describes relation between curvature form and curvature tensor:
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| :<math>R(u,v)w=\Omega(u\wedge v)w. </math>
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| This approach builds in all symmetries of curvature tensor except the ''first Bianchi identity'', which takes form
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| :<math>\Omega\wedge\theta=0</math>
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| where <math>\theta=\theta^i</math> is an ''n''-vector of 1-forms defined by <math>\theta^i(v)=\langle e_i,v\rangle</math>.
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| The ''second Bianchi identity'' takes form
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| :<math>D\Omega=0</math>
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| ''D'' denotes the [[exterior covariant derivative]]
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| ===The curvature operator===
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| It is sometimes convenient to think about curvature as an [[operator (mathematics)|operator]] <math>Q</math>
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| on tangent [[Outer product|bivector]]s (elements of <math>\Lambda^2(T)</math>), which is uniquely defined by the following identity:
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| :<math>\langle Q (u\wedge v),w\wedge z\rangle=\langle R(u,v)z,w \rangle.</math>
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| It is possible to do this precisely because of the symmetries of the curvature tensor (namely antisymmetry in the first and last pairs of indices, and block-symmetry of those pairs).
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| ==Further curvature tensors==
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| In general the following tensors and functions do not describe the curvature tensor completely,
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| however they play an important role.
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| === Scalar curvature ===
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| {{Main|Scalar curvature}}
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| Scalar curvature is a function on any Riemannian manifold, usually denoted by ''Sc''.
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| It is the full [[trace (linear algebra)|trace]] of the curvature tensor; given an [[orthonormal basis]]
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| <math>\{e_i\}</math> in the tangent space at ''p'' we have
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| :<math>S\! c=\sum_{i,j}\langle R(e_i,e_j)e_j,e_i\rangle=\sum_{i}\langle \text{Ric}(e_i),e_i\rangle, </math>
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| where ''Ric'' denotes [[Ricci tensor]]. The result does not depend on the choice of orthonormal basis. Starting with dimension 3, scalar curvature does not describe the curvature tensor completely.
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| ===Ricci curvature===
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| {{Main|Ricci curvature}}
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| Ricci curvature is a linear operator on tangent space at a point, usually denoted by ''Ric''.
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| Given an orthonormal basis
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| <math>\{e_i\}</math> in the tangent space at ''p'' we have
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| :<math>Ric(u)=\sum_{i} R(u,e_i)e_i.^{}_{} </math>
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| The result does not depend on the choice of orthonormal basis.
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| With four or more dimensions, Ricci curvature does not describe the curvature tensor completely.
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| Explicit expressions for the [[Ricci tensor]] in terms of the [[Levi-Civita connection]] is given in the article on [[Christoffel symbols]].
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| ===Weyl curvature tensor===
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| {{Main|Weyl tensor}}
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| The '''Weyl curvature tensor''' has the same symmetries as the curvature tensor, plus one extra: its Ricci curvature must vanish.
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| In dimensions 2 and 3 Weyl curvature vanishes, but if the dimension ''n'' > 3 then the second part can be non-zero.
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| *The curvature tensor can be decomposed into the part which depends on the Ricci curvature, and the Weyl tensor.
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| *If ''g′=fg'' for some positive scalar function ''f'' — a [[conformal map|conformal]] change of metric — then ''W ′ = W''.
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| *For a [[manifold]] of [[constant curvature]], the Weyl tensor is zero.
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| **Moreover, ''W=0'' if and only if the metric is locally [[conformal map|conformal]] to the standard Euclidean metric (equal to ''fg'', where ''g'' is the standard metric in some coordinate frame and ''f'' is some scalar function).
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| ===Ricci decomposition===
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| {{main|Ricci decomposition}}
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| Although individually, the Weyl tensor and Ricci tensor do not in general determine the full curvature tensor, the Riemann curvature tensor can be decomposed into a Weyl part and a Ricci part. This decomposition is known as the Ricci decomposition, and plays an important role in the [[conformal geometry]] of Riemannian manifolds. In particular, it can be used to show that if the metric is rescaled by a conformal factor of <math>e^{2f}</math>, then the Riemann curvature tensor changes to (seen as a (0, 4)-tensor):
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| : <math>e^{2f}\left(R+\left(\text{Hess}(f)-df\otimes df+\frac{1}{2}\|\text{grad}(f)\|^2 g\right) {~\wedge\!\!\!\!\!\!\bigcirc~} g\right)</math>
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| where <math>{~\wedge\!\!\!\!\!\!\bigcirc~}</math> denotes the [[Kulkarni–Nomizu product]] and Hess is the Hessian.
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| ==Calculation of curvature==
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| For calculation of curvature | |
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| *of hypersurfaces and submanifolds see [[second fundamental form]],
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| *in coordinates see the [[list of formulas in Riemannian geometry]] or [[covariant derivative]],
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| *by moving frames see [[Cartan connection]] and [[curvature form]].
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| *the [[Jacobi equation]] can help if one knows something about the behavior of [[geodesic#Riemannian and pseudo-Riemannian manifolds|geodesic]]s.
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| ==References==
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| *{{cite book | author=Kobayashi, Shoshichi and Nomizu, Katsumi | title = [[Foundations of Differential Geometry]], Vol. 1 | publisher=Wiley-Interscience | year=1996 (New edition) |isbn = 0-471-15733-3}}
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| == Notes ==
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| <references/>
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| {{curvature}}
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| [[Category:Riemannian manifolds]]
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| [[Category:Curvature (mathematics)]]
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