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((Short description|2-tensor obtained as a contraction of the Rieman curvature 4-tensor on a Riemannian manifold))
In [[differential geometry]], the '''Ricci curvature tensor''', named after [[Gregorio Ricci-Curbastro]], is a geometric object which is determined by a choice of [[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian manifold|pseudo-Riemannian metric]] on a [[manifold]]. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary [[Euclidean space]] or [[pseudo-Euclidean space]].

The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along [[geodesic]]s in the space. In [[general relativity]], which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the [[Raychaudhuri equation]]. Partly for this reason, the [[Einstein field equations]] propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe.

Like the metric tensor, the Ricci tensor assigns to each [[tangent space]] of the manifold a [[symmetric bilinear form]] ((harv|Besse|1987|p=43)).<ref>Here it is assumed that the manifold carries its unique [[Levi-Civita connection]]. For a general [[affine connection]], the Ricci tensor need not be symmetric.</ref> Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of the [[Laplace operator|Laplacian]] in the analysis of functions; in this analogy, the [[Riemann curvature tensor]], of which the Ricci curvature is a natural by-product, would correspond to the full matrix of second derivatives of a function. However, there are [[Laplace–Beltrami operator|other ways]] to draw the same analogy.

In [[Low-dimensional topology|three-dimensional topology]], the Ricci tensor contains all of the information which in higher dimensions is encoded by the more complicated [[Riemann curvature tensor]]. In part, this simplicity allows for the application of many geometric and analytic tools, which led to the [[solution of the Poincaré conjecture]] through the work of [[Richard S. Hamilton]] and [[Grigory Perelman]].

In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. [[comparison theorem]]) with the geometry of a constant curvature [[space form]]. This is since lower bounds on the Ricci tensor can be successfully used in studying the length functional in Riemannian geometry, as first shown in 1941 via [[Myers's theorem]].

One common source of the Ricci tensor is that it arises whenever one commutes the covariant derivative with the tensor Laplacian. This, for instance, explains its presence in the [[Bochner's formula|Bochner formula]], which is used ubiquitously in Riemannian geometry. For example, this formula explains why the gradient estimates due to [[Shing-Tung Yau]] (and their developments such as the Cheng-Yau and Li-Yau inequalities) nearly always depend on a lower bound for the Ricci curvature.

In 2007, [[John Lott (mathematician)|John Lott]], [[Karl-Theodor Sturm]], and [[Cedric Villani]] demonstrated decisively that lower bounds on Ricci curvature can be understood entirely in terms of the metric space structure of a Riemannian manifold, together with its volume form.<ref>((cite arXiv|last1=Lott|first1=John|last2=Villani|first2=Cedric|date=2006-06-23|title=Ricci curvature for metric-measure spaces via optimal transport|eprint=math/0412127))</ref> This established a deep link between Ricci curvature and [[Wasserstein metric|Wasserstein geometry]] and [[Transportation theory (mathematics)|optimal transport]], which is presently the subject of much research.((Citation needed|reason=Needs citation both to 2007 paper of Lott, Sturm and Villani, as well as citations to the "much research" which has come since. |date=January 2021))

==Definition==
Suppose that <math>\left( M, g \right)</math> is an <math>n</math>-dimensional 
[[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian manifold]], equipped 
with its [[Levi-Civita connection]] <math>\nabla</math>. The 
[[Riemannian curvature tensor|Riemann curvature]] of <math>M</math> is a map which
takes smooth vector fields <math>X</math>, <math>Y</math>, and <math>Z</math>, 
and returns the vector field<math display="block">R(X,Y)Z := \nabla_X\nabla_Y Z - \nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z</math>on [[vector field]]s <math>X, Y, Z</math>. Since <math>R</math> is a tensor field, for 
each point <math>p \in M</math>, it gives rise to a (multilinear) map:<math display="block">\operatorname{R}_p:T_pM\times T_pM\times T_pM\to T_pM.</math>Define for each point <math>p \in M</math> the map <math>\operatorname{Ric}_p:T_pM\times T_pM\to\mathbb{R}</math> by 

<math display="block">\operatorname{Ric}_p(Y,Z) := \operatorname{tr}\big(X\mapsto \operatorname{R}_p(X,Y)Z\big).</math>

That is, having fixed <math>Y</math> and <math>Z</math>, then for any orthonormal basis 
<math>v_1, \ldots, v_n</math> of the vector space <math>T_p M</math>, one has

<math display="block">\operatorname{Ric}_p(Y,Z) = \sum_{i=1} \langle\operatorname{R}_p(v_i, Y) Z, v_i \rangle.</math>

It is a standard exercise of (multi)linear
algebra to verify that this definition does not depend on the choice of the basis 
<math>v_1, \ldots, v_n</math>.

In [[abstract index notation]], 

<math display="block">\mathrm{Ric}_{ab} = \mathrm{R}^{c}{}_{bca} = \mathrm{R}^{c}{}_{acb}. </math>

'''Sign conventions.''' Note that some sources define <math>R(X,Y)Z</math> to be 
what would here be called <math>-R(X,Y)Z;</math> they would then define 
<math>\operatorname{Ric}_p</math> as <math>-\operatorname{tr}(X\mapsto \operatorname{R}_p(X,Y)Z).</math> 
Although sign conventions differ about the Riemann tensor, they do not differ about 
the Ricci tensor.

===Definition via local coordinates on a smooth manifold===
Let <math>\left( M, g \right)</math> be a smooth [[Riemannian manifold|Riemannian]]
or [[Pseudo-Riemannian manifold|pseudo-Riemannian]] <math>n</math>-manifold. 
Given a smooth chart <math>\left( U, \varphi \right)</math> one then has functions
<math>g_{ij}: \varphi(U) \rightarrow \mathbb{R}</math> and 
<math>g^{ij}: \varphi(U) \rightarrow \mathbb{R}</math> for each 
<math>i, j = 1, \ldots, n</math> which satisfy

<math display="block">
\sum_{k=1}^n g^{ik}(x)g_{kj}(x) = \delta^{i}_j = \begin{cases} 1 & i=j \\ 0 & i \neq j \end{cases}
</math>

for all <math>x \in \varphi(U)</math>. The latter shows that, expressed as
matrices, <math>g^{ij}(x) = (g^{-1})_{ij}(x)</math>.
The functions <math>g_{ij}</math> are defined by evaluating <math>g</math> on 
coordinate vector fields, while the functions <math>g^{ij}</math> are defined so
that, as a matrix-valued function, they provide an inverse to the matrix-valued 
function <math>x \mapsto g_{ij}(x)</math>.

Now define, for each <math>a</math>, <math>b</math>, <math>c</math>, <math>i</math>, 
and <math>j</math> between 1 and <math>n</math>, the functions

<math display="block">\begin{align}
  \Gamma_{ab}^c &:= \frac{1}{2} \sum_{d=1}^n \left(\frac{\partial g_{bd)){\partial x^a} + \frac{\partial g_{ad)){\partial x^b} - \frac{\partial g_{ab)){\partial x^d}\right)g^{cd}\\
  R_{ij} &:= \sum_{a=1}^n\frac{\partial\Gamma_{ij}^a}{\partial x^a} - \sum_{a=1}^n\frac{\partial\Gamma_{ai}^a}{\partial x^j} + \sum_{a=1}^n\sum_{b=1}^n\left(\Gamma_{ab}^a\Gamma_{ij}^b - \Gamma_{ib}^a\Gamma_{aj}^b\right)
\end{align}</math>

as maps <math>\varphi: U \rightarrow \mathbb{R}</math>.

Now let <math>\left( U, \varphi \right)</math> and <math>\left( V, \psi \right)</math> be two smooth charts with <math>U \cap V \neq \emptyset</math>.
Let <math>R_{ij}: \varphi(U) \rightarrow \mathbb{R}</math> be the functions computed as above via the chart <math>\left( U, \varphi \right)</math> and let <math>r_{ij}: \psi(V) \rightarrow \mathbb{R}</math> be the functions computed as above via the chart <math>\left( V, \psi \right)</math>.
Then one can check by a calculation with the chain rule and the product rule that

<math display="block">
R_{ij}(x) = \sum_{k,l=1}^n r_{kl}\left(\psi\circ\varphi^{-1}(x)\right)D_i\Big|_x \left(\psi\circ\varphi^{-1}\right)^kD_j\Big|_x \left(\psi\circ\varphi^{-1}\right)^l.
</math>

where <math>D_{i}</math> is the first derivative along <math>i</math>th direction
of <math>\mathbb{R}^n</math>.
This shows that the following definition does not depend on the choice of 
<math>\left( U, \varphi \right)</math>.
For any <math>p \in U</math>, define a bilinear map 
<math>\operatorname{Ric}_p : T_p M \times T_p M \rightarrow \mathbb{R}</math> by

<math display="block">
(X, Y) \in T_p M \times T_p M \mapsto \operatorname{Ric}_p(X,Y) = \sum_{i,j=1}^n R_{ij}(\varphi(x))X^i(p)Y^j(p),
</math>

where <math>X^1, \ldots, X^n</math> and <math>Y^1, \ldots, Y^n</math> are the 
components of the tangent vectors at <math>p</math> in <math>X</math> and <math>Y</math> relative to 
the coordinate vector fields of <math>\left( U, \varphi \right)</math>.

It is common to abbreviate the above formal presentation in the following style:
((block indent| em = 2 | text =
Let <math>M</math> be a smooth manifold, and let ((mvar|g)) be a Riemannian or pseudo-Riemannian metric. In local smooth coordinates, define the Christoffel symbols 

<math display="block">\begin{align}
  \Gamma_{ij}^k &:= \frac{1}{2}g^{kl}\left(\partial_ig_{jl} + \partial_jg_{il} - \partial_lg_{ij}\right)\\
  R_{jk} &:= \partial_i\Gamma_{jk}^i - \partial_j\Gamma_{ki}^i + \Gamma_{ip}^i\Gamma_{jk}^p - \Gamma_{jp}^i\Gamma_{ik}^p.
\end{align}</math>

It can be directly checked that

<math display="block">R_{jk} = \widetilde{R}_{ab}\frac{\partial\widetilde{x}^a}{\partial x^j}\frac{\partial\widetilde{x}^b}{\partial x^k},</math>

so that <math>R_{ij}</math> define a (0,2)-tensor field on <math>M</math>. In 
particular, if <math>X</math> and <math>Y</math> are vector fields on <math>M</math>,
then relative to any smooth coordinates one has

<math display="block">
\begin{align}
R_{jk}X^jY^k
  &= \left(\widetilde{R}_{ab}\frac{\partial\widetilde{x}^a}{\partial x^j}\frac{\partial\widetilde{x}^b}{\partial x^k}\right)\left(\widetilde{X}^c\frac{\partial x^j}{\partial\widetilde{x}^c}\right)\left(\widetilde{Y}^d\frac{\partial x^k}{\partial\widetilde{x}^d}\right) \\
  &= \widetilde{R}_{ab}\widetilde{X}^c\widetilde{Y}^d\left(\frac{\partial\widetilde{x}^a}{\partial x^j}\frac{\partial x^j}{\partial\widetilde{x}^c}\right)\left(\frac{\partial\widetilde{x}^b}{\partial x^k}\frac{\partial x^k}{\partial\widetilde{x}^d}\right) \\
  &= \widetilde{R}_{ab}\widetilde{X}^c\widetilde{Y}^d\delta_c^a\delta_d^b \\
  &= \widetilde{R}_{ab}\widetilde{X}^a\widetilde{Y}^b.
\end{align}
</math>
))

The final line includes the demonstration that the bilinear map Ric is well-defined, 
which is much easier to write out with the informal notation.
===Comparison of the definitions===
The two above definitions are identical. The formulas defining <math>\Gamma_{ij}^k</math> and <math>R_{ij}</math> in the coordinate approach have an exact parallel in the formulas defining the Levi-Civita connection, and the Riemann curvature via the Levi-Civita connection. Arguably, the definitions directly using local coordinates are preferable, since the "crucial property" of the Riemann tensor mentioned above requires <math>M</math> to be Hausdorff in order to hold. By contrast, the local coordinate approach only requires a smooth atlas. It is also somewhat easier to connect the "invariance" philosophy underlying the local approach with the methods of constructing more exotic geometric objects, such as [[spinor field]]s.

The complicated formula defining <math>R_{ij}</math> in the introductory section is the same as that in the following section. The only difference is that terms have been grouped so that it is easy to see that <math>R_{ij}=R_{ji}.</math>

==Properties==
As can be seen from the symmetries of the Riemann curvature tensor,  the Ricci tensor of a Riemannian 
manifold is [[symmetric tensor|symmetric]], in the sense that

<math display="block">\operatorname{Ric}(X ,Y) = \operatorname{Ric}(Y,X)</math>

for all <math>X,Y\in T_pM.</math> 

It thus follows linear-algebraically that the Ricci tensor is completely determined 
by knowing the quantity <math>\operatorname{Ric}(X, X)</math> for all vectors 
<math>X</math> of unit length. This function on the set of unit tangent vectors 
is often also called the '''Ricci curvature''', since knowing it is equivalent to 
knowing the Ricci curvature tensor.

The Ricci curvature is determined by the [[sectional curvature]]s of a Riemannian 
manifold, but generally contains less information. Indeed, if <math>\xi</math> is a 
vector of unit length on a Riemannian <math>n</math>-manifold, then 
<math>\operatorname{Ric}(\xi, \xi)</math> is precisely <math>(n - 1)</math>
times the average value of the sectional curvature, taken over all the 2-planes 
containing <math>\xi</math>.  There is an <math>(n - 2)</math>-dimensional family 
of such 2-planes, and so only in dimensions 2 and 3 does the Ricci tensor determine 
the full curvature tensor. A notable exception is when the manifold is given a 
priori as a [[hypersurface]] of [[Euclidean space]]. The [[second fundamental form]], 
which determines the full curvature via the [[Gauss–Codazzi equations|Gauss–Codazzi equation]], 
is itself determined by the Ricci tensor and the [[Principal curvature|principal directions]] 
of the hypersurface are also the [[Eigenvalues and eigenvectors|eigendirections]] of the Ricci tensor. The 
tensor was introduced by Ricci for this reason.

As can be seen from the second Bianchi identity, one has

<math display="block">\operatorname{div}\operatorname{Ric} = \frac{1}{2}dR,</math>

where <math>R</math> is the [[scalar curvature]], defined in local coordinates as <math>g^{ij}R_{ij}.</math> This is often called the contracted second Bianchi identity.

===Informal properties===
The Ricci curvature is sometimes thought of as (a negative multiple of) the 
[[Laplacian]] of the metric tensor ((harv|Chow|Knopf|2004|loc=Lemma 3.32)).<ref>((Cite book |last=Chow |first=Bennett |url=https://www.worldcat.org/oclc/54692148 |title=The Ricci flow : an introduction |date=2004 |publisher=American Mathematical Society |others=Dan Knopf |isbn=0-8218-3515-7 |location=Providence, R.I. |oclc=54692148))</ref>  Specifically, in [[harmonic coordinates|harmonic]] local coordinates the components satisfy

<math display="block">R_{ij} = -\frac{1}{2}\Delta \left(g_{ij}\right) + \text{lower-order terms},</math>

where <math>\Delta = \nabla \cdot \nabla</math> is the [[Laplace–Beltrami operator]],
here regarded  as acting on the locally-defined functions <math>g_{ij}</math>. 
This fact motivates, for instance, the introduction of the [[Ricci flow]] equation 
as a natural extension of the [[heat equation]] for the metric. Alternatively, 
in a [[normal coordinates|normal coordinate system]] based at <math>p</math>,

<math display="block">R_{ij} = -\frac{2}{3}\Delta \left(g_{ij}\right).</math>

==Direct geometric meaning==
Near any point <math>p</math> in a Riemannian manifold <math>\left( M, g \right)</math>,
one can define preferred local coordinates, called [[geodesic normal coordinates]].
These are adapted to the metric so that geodesics through <math>p</math> correspond
to straight lines through the origin, in  such a manner that the geodesic distance
from <math>p</math> corresponds to the  Euclidean distance from the origin. 
In these coordinates, the metric tensor is well-approximated by the Euclidean 
metric, in the precise sense that

<math display="block">g_{ij} = \delta_{ij} + O \left(|x|^2\right) .</math>

In fact, by taking the [[Taylor expansion]] of the metric applied to a [[Jacobi field]] along a radial geodesic in the normal coordinate system, one has
<math display="block">g_{ij} = \delta_{ij} - \frac{1}{3} R_{ikjl}x^kx^l + O\left(|x|^3\right) .</math>

In these coordinates, the metric [[volume element]] then has the following expansion at ((mvar|p)):

<math display="block">d\mu_g = \left[ 1 - \frac{1}{6} R_{jk}x^j x^k+ O\left(|x|^3\right) \right] d\mu_\text{Euclidean} ,</math>

which follows by expanding the square root of the [[determinant]] of the metric.

Thus, if the Ricci curvature <math>\operatorname{Ric}(\xi, \xi)</math> is positive
in the direction of a vector <math>\xi</math>, the conical region in <math>M</math> 
swept out by a tightly focused family of geodesic segments of length 
<math>\varepsilon</math> emanating from <math>p</math>, with initial velocity inside 
a small cone about <math>\xi</math>, will have smaller volume than the corresponding
conical region in Euclidean space, at least provided that <math>\varepsilon</math> 
is sufficiently small. Similarly, if the Ricci curvature is negative in the 
direction of a given vector <math>\xi</math>, such a conical region in the manifold 
will instead have larger volume than it would in Euclidean space.

The Ricci curvature is essentially an average of curvatures in the planes including
<math>\xi</math>. Thus if a cone emitted with an initially circular (or spherical)
cross-section becomes distorted into an ellipse ([[ellipsoid]]), it is possible
for the volume distortion to vanish if the distortions along the 
[[Principal axis theorem|principal axes]] counteract one another. The Ricci 
curvature would then vanish along <math>\xi</math>. In physical applications, the 
presence of a nonvanishing [[sectional curvature]] does not necessarily indicate the
presence of any mass locally; if an initially circular cross-section of a cone 
of [[worldline]]s later becomes elliptical, without changing its volume, then 
this is due to tidal effects from a mass at some other location.

== Applications ==
Ricci curvature plays an important role in [[general relativity]], where it is 
the key term in the [[Einstein field equations]].

Ricci curvature  also appears in  the [[Ricci flow]] equation, where certain 
one-parameter families of Riemannian metrics are singled out as solutions of a 
geometrically-defined partial differential equation. This system of equations 
can be thought of as a geometric analog of the [[heat equation]], and was first 
introduced by [[Richard S. Hamilton]] in 1982. Since heat tends to spread through 
a solid until the body reaches an equilibrium state of constant temperature, if 
one is given a manifold, the Ricci flow may be hoped to produce an 'equilibrium' 
Riemannian metric which is [[Einstein metric|Einstein]] or of constant curvature. 
However, such a clean "convergence" picture cannot be achieved since many manifolds 
cannot support such metrics. A detailed study of the nature of solutions of the 
Ricci flow, due principally to Hamilton and [[Grigori Perelman]], shows that the 
types of "singularities" that occur along a Ricci flow, corresponding to the 
failure of convergence, encodes deep information about 3-dimensional topology. 
The culmination of this work was a proof of the [[geometrization conjecture]] 
first proposed by [[William Thurston]] in the 1970s, which can be thought of as 
a classification of compact 3-manifolds.

On a [[Kähler manifold]], the  Ricci curvature determines the first [[Chern class]] 
of the manifold (mod torsion). However, the Ricci curvature has no analogous 
topological interpretation on a generic Riemannian manifold.

==Global geometry and topology==
Here is a short list of global results concerning manifolds with positive Ricci curvature; see also [[Riemannian geometry#Local to global theorems|classical theorems of Riemannian geometry]]. Briefly, positive Ricci curvature of a Riemannian manifold has strong topological consequences, while (for dimension at least 3), negative Ricci curvature has ''no'' topological implications. (The Ricci curvature is said to be '''positive''' if the Ricci curvature function <math>\operatorname{Ric}(\xi, \xi)</math> is positive on the set of non-zero tangent vectors <math>\xi</math>.)  Some results are also known for pseudo-Riemannian manifolds.

#[[Myers theorem|Myers' theorem]] (1941) states that if the Ricci curvature is bounded from below on a complete Riemannian ''n''-manifold by <math>(n - 1)k > 0</math>, then the manifold has diameter <math>\leq \pi / \sqrt{k}</math>. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite [[fundamental group]]. [[Shiu-Yuen Cheng|Cheng]] (1975) showed that, in this setting, equality in the diameter inequality occurs if only if the manifold is [[Isometry|isometric]] to a sphere of a constant curvature <math>k</math>.
#The [[Bishop–Gromov inequality]] states that if a complete <math>n</math>-dimensional Riemannian manifold has non-negative Ricci curvature, then the volume of a geodesic ball is less than or equal to the volume of a geodesic ball of the same radius in Euclidean <math>n</math>-space. Moreover, if <math>v_p(R)</math> denotes the volume of the ball with center <math>p</math> and radius <math>R</math> in the manifold and <math>V(R) = c_n R^n</math> denotes the volume of the ball of radius <math>R</math> in Euclidean <math>n</math>-space then the function <math>v_p(R) / V(R)</math> is nonincreasing. This can be generalized to any lower bound on the Ricci curvature (not just nonnegativity), and is the key point in the proof of [[Gromov's compactness theorem (geometry)|Gromov's compactness theorem]].)
#The Cheeger–Gromoll [[splitting theorem]] states that if a complete Riemannian manifold <math>\left( M, g \right)</math> with <math>\operatorname{Ric} \geq 0</math> contains a ''line'', meaning a geodesic <math>\gamma : \mathbb{R} \to M</math>  such that <math>d(\gamma(u), \gamma(v)) = \left| u - v \right|</math> for all <math>u, v \in \mathbb{R}</math>, then it is isometric to a product space <math>\mathbb{R} \times L</math>. Consequently, a complete manifold of positive Ricci curvature can have at most one topological end.  The theorem is also true  under some additional hypotheses for complete [[Lorentzian manifold]]s (of metric  signature <math>\left( + - - \ldots \right)</math>) with non-negative Ricci tensor (((harvnb|Galloway|2000))).
#Hamilton's first [[Ricci flow|convergence theorem]] for Ricci flow has, as a corollary, that the only compact 3-manifolds which have Riemannian metrics of  positive Ricci curvature are the quotients of the 3-sphere by discrete subgroups  of SO(4) which act properly discontinuously. He later extended this to allow for  nonnegative Ricci curvature. In particular, the only simply-connected possibility  is the 3-sphere itself.
These results, particularly Myers' and Hamilton's, show that positive Ricci curvature has strong topological consequences. By contrast, excluding the case of surfaces, negative Ricci curvature is now known to have ''no'' topological implications; ((harvtxt|Lohkamp|1994)) has shown that any manifold of dimension greater than two admits a complete Riemannian metric of negative Ricci curvature. In the case of two-dimensional manifolds, negativity of the Ricci curvature is synonymous with negativity of the Gaussian curvature, which has very clear [[Gauss-Bonnet theorem|topological implications]]. There are very few two-dimensional manifolds which fail to admit Riemannian metrics of negative Gaussian curvature.
== Behavior under conformal rescaling ==
If the metric <math>g</math> is changed by multiplying it by a conformal factor 
<math>e^{2f}</math>, the Ricci tensor of the new, conformally-related metric 
<math>\tilde{g} = e^{2f} g</math> is given  ((harv|Besse|1987|p=59)) by

<!--PLEASE DON'T CHANGE \Delta TO -\Delta.  The sign is correct for our Laplacian convention (and agrees with the cited source).-->
<math display="block">\widetilde{\operatorname{Ric))=\operatorname{Ric}+(2-n)\left[ \nabla df-df\otimes df\right]+\left[\Delta f -(n-2)\|df\|^2\right]g ,</math>

where <math>\Delta = *d*d</math> is the (positive spectrum) Hodge Laplacian, i.e., 
the ''opposite'' of the usual trace of the Hessian.

In particular, given a point <math>p</math> in a Riemannian manifold, it is always 
possible to find metrics conformal to the given metric <math>g</math> for which the 
Ricci tensor vanishes at <math>p</math>. Note, however, that this is only pointwise 
assertion; it is usually impossible to make the Ricci curvature vanish identically 
on the entire manifold by a conformal rescaling.

For two dimensional manifolds, the above formula shows that if <math>f</math> is a 
[[harmonic function]], then the conformal scaling <math>g \mapsto e^{2f}g</math> 
does not change the Ricci tensor (although it still changes its trace with respect 
to the metric unless <math>f = 0</math>.

==Trace-free Ricci tensor==
In [[Riemannian geometry]] and [[pseudo-Riemannian geometry]], the 
'''trace-free Ricci tensor''' (also called '''traceless Ricci tensor''') of a 
Riemannian or pseudo-Riemannian <math>n</math>-manifold <math>\left( M, g \right)</math> 
is the tensor defined by

<math display="block">Z = \operatorname{Ric} - \frac{1}{n}Rg ,</math>

where <math>\operatorname{Ric}</math> and <math>R</math> denote the Ricci curvature 
and [[scalar curvature]] of <math>g</math>. The name of this object reflects the 
fact that its [[Trace (linear algebra)|trace]] automatically vanishes: 
<math>\operatorname{tr}_gZ\equiv g^{ab}Z_{ab} = 0.</math> However, it is quite an 
important tensor since it reflects an "orthogonal decomposition" of the Ricci tensor.

=== The orthogonal decomposition of the Ricci tensor ===
The following, not so trivial, property is

<math display="block">\operatorname{Ric} = Z + \frac{1}{n}Rg.</math>

It is less immediately obvious that the two terms on the right hand side are orthogonal 
to each other:

<math display="block">\left\langle Z, \frac{1}{n}Rg\right\rangle_g \equiv g^{ab}\left(R_{ab} - \frac{1}{n}Rg_{ab}\right) = 0.</math>

An identity which is intimately connected with this (but which could be proved directly) 
is that

<math display="block">\left|\operatorname{Ric}\right|_g^2 = |Z|_g^2 + \frac{1}{n}R^2.</math>

===The trace-free Ricci tensor and Einstein metrics===
By taking a divergence, and using the contracted Bianchi identity, one sees that 
<math>Z = 0</math> implies <math display="inline">\frac{1}{2}dR - \frac{1}{n}dR = 0</math>. 
So, provided that ((math|''n'' ≥ 3)) and <math>M</math> is connected, the vanishing 
of <math>Z</math> implies that the scalar curvature is constant. One can then see 
that the following are equivalent:

* <math>Z = 0</math>
* <math>\operatorname{Ric} = \lambda g</math> for some number <math>\lambda</math>
* <math>\operatorname{Ric} = \frac{1}{n}Rg</math>

In the Riemannian setting, the above orthogonal decomposition shows that 
<math>R^2 = n|\operatorname{Ric}|^2</math> is also equivalent to these conditions. 
In the pseudo-Riemmannian setting, by contrast, the condition <math>|Z|_g^2 = 0</math> 
does not necessarily imply <math>Z = 0,</math> so the most that one can say is that 
these conditions imply <math>R^2 = n \left|\operatorname{Ric}\right|_g^2.</math>

In particular, the vanishing of trace-free Ricci tensor characterizes 
[[Einstein manifold]]s, as defined by the condition <math>\operatorname{Ric} = \lambda g</math> 
for a number <math>\lambda.</math> In [[general relativity]], this equation states 
that <math>\left( M, g \right)</math> is a solution of Einstein's vacuum field 
equations with [[cosmological constant]].

==Kähler manifolds==
On a [[Kähler manifold]] <math>X</math>, the Ricci curvature determines the 
[[curvature form]] of the [[canonical bundle|canonical line bundle]] 
((harv|Moroianu|2007|loc=Chapter 12)).  The canonical line bundle is the top 
[[exterior power]] of the bundle of holomorphic [[Kähler differential]]s:

<math display="block">
\kappa = {\textstyle\bigwedge}^n ~ \Omega_X.
</math>

The Levi-Civita connection corresponding to the metric on <math>X</math> gives 
rise to a connection on <math>\kappa</math>. The curvature of this connection is 
the 2-form defined by 

<math display="block">\rho(X,Y) \;\stackrel{\text{def)){=}\; \operatorname{Ric}(JX,Y)</math>

where <math>J</math> is the [[complex manifold|complex structure]] map on the 
tangent bundle determined by the structure of the Kähler manifold. The Ricci 
form is a [[closed and exact forms|closed]] 2-form. Its [[cohomology class]] is, 
up to a real constant factor, the first [[Chern class]] of the canonical bundle, 
and is therefore a topological invariant of <math>X</math> (for compact <math>X</math>) 
in the sense that it depends only on the topology of <math>X</math> and the 
[[homotopy class]] of the complex structure.

Conversely, the Ricci form determines the Ricci tensor by

<math display="block">\operatorname{Ric}(X, Y) = \rho(X, JY).</math>

In local holomorphic coordinates <math>z^\alpha</math>, the Ricci form is given by

<math display="block">\rho = -i\partial\overline{\partial}\log\det\left(g_{\alpha\overline{\beta))\right)</math>

where ((math|∂)) is the [[Dolbeault operator]] and

<math display="block">g_{\alpha\overline{\beta)) = g\left(\frac{\partial}{\partial z^\alpha}, \frac{\partial}{\partial\overline{z}^\beta}\right).</math>

If the Ricci tensor vanishes, then the canonical bundle is flat, so the 
[[G-structure|structure group]] can be locally reduced to a subgroup of the 
special linear group <math>SL(n; \mathbb{C})</math>.  However, Kähler manifolds 
already possess [[holonomy]] in <math>U(n)</math>, and so the (restricted) 
holonomy of a Ricci-flat Kähler manifold is contained in <math>SU(n)</math>. 
Conversely, if the (restricted) holonomy of a 2<math>n</math>-dimensional Riemannian 
manifold is contained in <math>SU(n)</math>, then the manifold is a Ricci-flat 
Kähler manifold ((harv|Kobayashi|Nomizu|1996|loc=IX, §4)).

==Generalization to affine connections==
The Ricci tensor can also be generalized to arbitrary [[affine connection]]s, 
where it is an invariant that plays an especially important role in the study of 
[[projective differential geometry|projective geometry]] (geometry associated to 
unparameterized geodesics) ((harv|Nomizu|Sasaki|1994)).  If <math>\nabla</math> 
denotes an affine connection, then the curvature tensor <math>R</math> is the 
(1,3)-tensor defined by

<math display="block">R(X,Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z</math>

for any vector fields <math>X, Y, Z</math>. The Ricci tensor is defined to be the trace:

<math display="block">\operatorname{ric}(X,Y) = \operatorname{tr}\big(Z\mapsto R(Z,X)Y\big).</math>

In this more general situation, the Ricci tensor is symmetric if and only if there 
exists locally a parallel [[volume form]] for the connection.

== Discrete Ricci curvature ==
Notions of Ricci curvature on discrete manifolds have been defined on graphs and
networks, where they quantify local divergence properties of edges. Ollivier's 
Ricci curvature is defined using optimal transport theory.<ref>((Cite journal |last=Ollivier |first=Yann |date=2009-02-01 |title=Ricci curvature of Markov chains on metric spaces |journal=Journal of Functional Analysis |language=en |volume=256 |issue=3 |pages=810–864 |doi=10.1016/j.jfa.2008.11.001 |s2cid=14316364 |issn=0022-1236|doi-access=free ))</ref> 
A different (and earlier) notion, Forman's Ricci curvature, is based on 
topological arguments.<ref>((Cite journal |last=Forman |date=2003-02-01 |title=Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature |journal=Discrete & Computational Geometry |language=en |volume=29 |issue=3 |pages=323–374 |doi=10.1007/s00454-002-0743-x |s2cid=9584267 |issn=1432-0444|doi-access=free ))</ref>

==See also==
((cols))
*[[Curvature of Riemannian manifolds]]
*[[Scalar curvature]]
*[[Ricci calculus]]
*[[Ricci decomposition]]
*[[Ricci-flat manifold]]
*[[Christoffel symbols]]
*[[Introduction to the mathematics of general relativity]]
((colend))

== Footnotes ==
((reflist))

==References==
*((citation|first=A.L.|last=Besse|author-link=Arthur Besse|title=Einstein manifolds|publisher=Springer|year=1987|isbn=978-3-540-15279-8)).
*((Citation |last1=Chow |first1=Bennet  |last2=Knopf |first2=Dan  |name-list-style=amp | title =The Ricci Flow: an introduction| publisher=American Mathematical Society | year=2004| isbn=0-8218-3515-7)).
*((citation|first=L.P.|last=Eisenhart|author-link=Luther P. Eisenhart|title=Riemannian geometry|publisher=Princeton Univ. Press|year=1949)).
*Forman (2003), "Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature", ''Discrete & Computational Geometry'', '''29''' (3): 323–374. [[doi:10.1007/s00454-002-0743-x]]. ISSN 1432-0444
*((citation|first=Gregory|last=Galloway|title=Maximum Principles for Null Hypersurfaces and Null Splitting Theorems|journal=Annales de l'Institut Henri Poincaré A|volume=1|year=2000|issue=3|pages=543–567|arxiv=math/9909158|bibcode = 2000AnHP....1..543G|doi=10.1007/s000230050006|s2cid=9619157)).
*((citation|first1=S.|last1=Kobayashi|first2=K.|last2=Nomizu|title=[[Foundations of Differential Geometry|Foundations of Differential Geometry, Volume 1]]|publisher=Interscience|year=1963)).
*((Citation | last1=Kobayashi | first1=Shoshichi | last2=Nomizu | first2=Katsumi | title=Foundations of Differential Geometry, Vol. 2 | publisher=[[Wiley-Interscience]] | isbn=978-0-471-15732-8 | year=1996)).
*((Citation | last1=Lohkamp | first1=Joachim | title=Metrics of negative Ricci curvature |mr=1307899 | year=1994 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=140 | issue=3 | pages=655–683 | doi=10.2307/2118620 | publisher=Annals of Mathematics | jstor=2118620)).
*((Citation | last1=Moroianu | first1=Andrei | title=Lectures on Kähler geometry | publisher=[[Cambridge University Press]] | series=London Mathematical Society Student Texts | isbn=978-0-521-68897-0 |mr=2325093 | year=2007 | volume=69 | doi=10.1017/CBO9780511618666| arxiv=math/0402223 | s2cid=209824092 ))
*((citation|first1=Katsumi|last1=Nomizu|first2=Takeshi|author-link=Katsumi Nomizu|last2=Sasaki|title=Affine differential geometry|year=1994|publisher=Cambridge University Press|isbn=978-0-521-44177-3|url-access=registration|url=https://archive.org/details/affinedifferenti0000nomi)).
*Ollivier, Yann (2009), "Ricci curvature of Markov chains on metric spaces", ''Journal of Functional Analysis'' '''256''' (3): 810–864. [[doi:10.1016/j.jfa.2008.11.001]]. ISSN 0022-1236
*((citation|first=G.|last=Ricci|author-link=Gregorio Ricci-Curbastro| title=Direzioni e invarianti principali in una varietà qualunque |journal=Atti R. Inst. Veneto|volume=63 |issue=2 |year=1903–1904|pages=1233–1239)).
*((springer|id=r/r081800|title=Ricci tensor|author=L.A. Sidorov))
*((springer|id=R/r081780|title=Ricci curvature|author=L.A. Sidorov))
*Najman, Laurent and Romon, Pascal (2017): Modern approaches to discrete curvature, Springer (Cham), Lecture notes in mathematics

==External links==
*Z. Shen, [[Christina Sormani|C. Sormani]] [https://arxiv.org/abs/math/0606774 "The Topology of Open Manifolds with Nonnegative Ricci Curvature"] (a survey)
*G. Wei, [https://arxiv.org/abs/math/0612107 "Manifolds with A Lower Ricci Curvature Bound"] (a survey)

((curvature))
((Riemannian geometry))
((Manifolds))
((tensors))

[[Category:Curvature (mathematics)]]
[[Category:Differential geometry]]
[[Category:Riemannian geometry]]
[[Category:Riemannian manifolds]]
[[Category:Tensors in general relativity]]

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