In
differential geometry, the Ricci curvature tensor, named after
Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus.
With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on t ...
, is a geometric object which is determined by a choice of
Riemannian or
pseudo-Riemannian metric on a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
. 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
or
pseudo-Euclidean space In mathematics and theoretical physics, a pseudo-Euclidean space is a finite- dimensional real -space together with a non-degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving
q(x ...
.
The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along
geodesics in the space. In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
, 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
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
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 . Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is ...
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
other ways to draw the same analogy.
In
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
Richard Streit Hamilton (born 10 January 1943) is an American mathematician who serves as the Davies Professor of Mathematics at Columbia University. He is known for contributions to geometric analysis and partial differential equations. 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 formula
In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold (M, g) to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.
Formal statement
If u \colon M \ri ...
, 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,
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. This established a deep link between Ricci curvature and
Wasserstein geometry and
optimal transport, which is presently the subject of much research.
Definition
Suppose that
is an
-dimensional
Riemannian or
pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which t ...
, equipped
with its
Levi-Civita connection
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
. The
Riemann curvature of
is a map which
takes smooth vector fields
,
, and
,
and returns the vector field
on
vector fields
. The crucial property of this mapping
is that if
and
are smooth vector
fields such that
and
define the same element of
some tangent space
, and
and
also
define the same element of
, and
and
also define the same element of
, then the vector fields
and
also define the same element
of
.
The implication is that the Riemann curvature, which is a priori a mapping with
vector field inputs and a vector field output, can actually be viewed as a mapping
with tangent vector inputs and a tangent vector output. That is, it defines for
each point
a (multilinear) map:
Define for each point
the map
by
That is, having fixed
and
, then for any basis
of the vector space
, one defines
where for any fixed
, the numbers
are the coordinates of
relative to
the basis
. It is a standard exercise of (multi)linear
algebra to verify that this definition does not depend on the choice of the basis
.
Sign conventions. Note that some sources define
to be
what would here be called
they would then define
as
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
be a smooth
Riemannian
or
pseudo-Riemannian -manifold.
Given a smooth chart
one then has functions
and
for each
which satisfy
for all
. The latter shows that, expressed as
matrices,
.
The functions
are defined by evaluating
on
coordinate vector fields, while the functions
are defined so
that, as a matrix-valued function, they provide an inverse to the matrix-valued
function
.
Now define, for each
,
,
,
,
and
between 1 and
, the functions
as maps
.
Now let
and
be two smooth charts with
.
Let
be the functions computed as above via the chart
and let
be the functions computed as above via the chart
.
Then one can check by a calculation with the chain rule and the product rule that
where
is the first derivative along
th direction
of
.
This shows that the following definition does not depend on the choice of
.
For any
, define a bilinear map
by
where
and
are the
components of the tangent vectors at
in
and
relative to
the coordinate vector fields of
.
It is common to abbreviate the above formal presentation in the following style:
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
and
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
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 fields.
The complicated formula defining
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
Properties
As can be seen from the
Bianchi identities, the Ricci tensor of a Riemannian
manifold is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
, in the sense that
for all
It thus follows linear-algebraically that the Ricci tensor is completely determined
by knowing the quantity
for all vectors
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 curvatures of a Riemannian
manifold, but generally contains less information. Indeed, if
is a
vector of unit length on a Riemannian
-manifold, then
is precisely
times the average value of the sectional curvature, taken over all the 2-planes
containing
. There is an
-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
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Eucl ...
of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
. The
second fundamental form,
which determines the full curvature via the
Gauss–Codazzi equation,
is itself determined by the Ricci tensor and the
principal directions
of the hypersurface are also the
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
where
is the
scalar curvature, defined in local coordinates as
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
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is ...
of the metric tensor . Specifically, in
harmonic local coordinates the components satisfy
where
is the
Laplace–Beltrami operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named ...
,
here regarded as acting on the locally-defined functions
.
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 coordinate system based at
,
Direct geometric meaning
Near any point
in a Riemannian manifold
,
one can define preferred local coordinates, called
geodesic normal coordinates
In differential geometry, normal coordinates at a point ''p'' in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of ''p'' obtained by applying the exponential map to the tan ...
.
These are adapted to the metric so that geodesics through
correspond
to straight lines through the origin, in such a manner that the geodesic distance
from
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
In fact, by taking the
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
of the metric applied to a
Jacobi field along a radial geodesic in the normal coordinate system, one has
In these coordinates, the metric
volume element then has the following expansion at :
which follows by expanding the square root of the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
of the metric.
Thus, if the Ricci curvature
is positive
in the direction of a vector
, the conical region in
swept out by a tightly focused family of geodesic segments of length
emanating from
, with initial velocity inside
a small cone about
, will have smaller volume than the corresponding
conical region in Euclidean space, at least provided that
is sufficiently small. Similarly, if the Ricci curvature is negative in the
direction of a given vector
, 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
. Thus if a cone emitted with an initially circular (or spherical)
cross-section becomes distorted into an ellipse (
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
), it is possible
for the volume distortion to vanish if the distortions along the
principal axes counteract one another. The Ricci
curvature would then vanish along
. 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
worldlines 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
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
, where it is
the key term in the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
.
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
Richard Streit Hamilton (born 10 January 1943) is an American mathematician who serves as the Davies Professor of Mathematics at Columbia University. He is known for contributions to geometric analysis and partial differential equations. 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 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
Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
, 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
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
is positive on the set of non-zero tangent vectors
.) Some results are also known for pseudo-Riemannian manifolds.
#
Myers' theorem (1941) states that if the Ricci curvature is bounded from below on a complete Riemannian ''n''-manifold by
, then the manifold has diameter
. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite
fundamental group.
Cheng (1975) showed that, in this setting, equality in the diameter inequality occurs if only if the manifold is
isometric
The term ''isometric'' comes from the Greek for "having equal measurement".
isometric may mean:
* Cubic crystal system, also called isometric crystal system
* Isometre, a rhythmic technique in music.
* "Isometric (Intro)", a song by Madeon from ...
to a sphere of a constant curvature
.
#The
Bishop–Gromov inequality states that if a complete
-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
-space. Moreover, if
denotes the volume of the ball with center
and radius
in the manifold and
denotes the volume of the ball of radius
in Euclidean
-space then the function
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.)
#The Cheeger–Gromoll
splitting theorem states that if a complete Riemannian manifold
with
contains a ''line'', meaning a geodesic
such that
for all
, then it is isometric to a product space
. 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 manifolds (of metric signature
) with non-negative Ricci tensor ().
#Hamilton's first
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; 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
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
is changed by multiplying it by a conformal factor
, the Ricci tensor of the new, conformally-related metric
is given by
where
is the (positive spectrum) Hodge Laplacian, i.e.,
the ''opposite'' of the usual trace of the Hessian.
In particular, given a point
in a Riemannian manifold, it is always
possible to find metrics conformal to the given metric
for which the
Ricci tensor vanishes at
. 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
is a
harmonic function, then the conformal scaling
does not change the Ricci tensor (although it still changes its trace with respect
to the metric unless
.
Trace-free Ricci tensor
In
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to po ...
and
pseudo-Riemannian geometry, the
trace-free Ricci tensor (also called traceless Ricci tensor) of a
Riemannian or pseudo-Riemannian
-manifold
is the tensor defined by
where
and
denote the Ricci curvature
and
scalar curvature of
. The name of this object reflects the
fact that its
trace automatically vanishes:
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
It is less immediately obvious that the two terms on the right hand side are orthogonal
to each other:
An identity which is intimately connected with this (but which could be proved directly)
is that
The trace-free Ricci tensor and Einstein metrics
By taking a divergence, and using the contracted Bianchi identity, one sees that
implies
.
So, provided that and
is connected, the vanishing
of
implies that the scalar curvature is constant. One can then see
that the following are equivalent:
*
*
for some number
*
In the Riemannian setting, the above orthogonal decomposition shows that
is also equivalent to these conditions.
In the pseudo-Riemmannian setting, by contrast, the condition
does not necessarily imply
so the most that one can say is that
these conditions imply
In particular, the vanishing of trace-free Ricci tensor characterizes
Einstein manifold
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is e ...
s, as defined by the condition
for a number
In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
, this equation states
that
is a solution of Einstein's vacuum field
equations with
cosmological constant.
Kähler manifolds
On a
Kähler manifold , the Ricci curvature determines the
curvature form of the
canonical line bundle
. The canonical line bundle is the top
exterior power of the bundle of holomorphic
Kähler differentials:
The Levi-Civita connection corresponding to the metric on
gives
rise to a connection on
. The curvature of this connection is
the 2-form defined by
where
is the
complex structure map on the
tangent bundle determined by the structure of the Kähler manifold. The Ricci
form is a
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
(for compact
)
in the sense that it depends only on the topology of
and the
homotopy class
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
of the complex structure.
Conversely, the Ricci form determines the Ricci tensor by
In local holomorphic coordinates
, the Ricci form is given by
where is the
Dolbeault operator
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex number, complex coefficients.
Complex forms have broad applications in differential geometry. On comp ...
and
If the Ricci tensor vanishes, then the canonical bundle is flat, so the
structure group can be locally reduced to a subgroup of the
special linear group
. However, Kähler manifolds
already possess
holonomy in
, and so the (restricted)
holonomy of a Ricci-flat Kähler manifold is contained in
.
Conversely, if the (restricted) holonomy of a 2
-dimensional Riemannian
manifold is contained in
, then the manifold is a Ricci-flat
Kähler manifold .
Generalization to affine connections
The Ricci tensor can also be generalized to arbitrary
affine connections,
where it is an invariant that plays an especially important role in the study of
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...
(geometry associated to
unparameterized geodesics) . If
denotes an affine connection, then the curvature tensor
is the
(1,3)-tensor defined by
for any vector fields
. The Ricci tensor is defined to be the trace:
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.
A different (and earlier) notion, Forman's Ricci curvature, is based on
topological arguments.
See also
*
Curvature of Riemannian manifolds
*
Scalar curvature
*
Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to b ...
*
Ricci decomposition
*
Ricci-flat manifold
*
Christoffel symbols
*
Introduction to the mathematics of general relativity
Footnotes
References
*.
*.
*.
*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
*.
*.
*.
*.
*
*.
*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
*.
*
*
*Najman, Laurent and Romon, Pascal (2017): Modern approaches to discrete curvature, Springer (Cham), Lecture notes in mathematics
External links
*Z. Shen,
C. Sormanibr>
"The Topology of Open Manifolds with Nonnegative Ricci Curvature"(a survey)
*G. Wei
"Manifolds with A Lower Ricci Curvature Bound"(a survey)
{{tensors
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Differential geometry
Riemannian geometry
Riemannian manifolds
Tensors in general relativity