In the mathematical field of
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 point ...
, the scalar curvature (or the Ricci scalar) is a measure of the
curvature of a Riemannian manifold. To each point on a
Riemannian manifold, it assigns a single
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of
partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the
differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
, the scalar curvature is twice the
Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
.
The definition of scalar curvature via partial derivatives is also valid in the more general setting of
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 the ...
s. This is significant 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 scalar curvature of a
Lorentzian metric is one of the key terms 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 ...
. Furthermore, this scalar curvature is the
Lagrangian density
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
for the
Einstein–Hilbert action, the
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s of which are the Einstein field equations in
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
.
The geometry of Riemannian metrics with positive scalar curvature has been widely studied. On noncompact spaces, this is the context of the
positive mass theorem proved by
Richard Schoen and
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
in the 1970s, and reproved soon after by
Edward Witten
Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...
with different techniques. Schoen and Yau, and independently
Mikhael Gromov and
Blaine Lawson, developed a number of fundamental results on the topology of
closed manifolds supporting metrics of positive scalar curvature. In combination with their results,
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 ...
's construction of
Ricci flow with surgery in 2003 provided a complete characterization of these topologies in the three-dimensional case.
Definition
Given a
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
, the scalar curvature ''S'' (commonly also ''R'', or ''Sc'') is defined as the
trace of the
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
tensor with respect to the metric:
:
The scalar curvature cannot be computed directly from the Ricci curvature since the latter is a (0,2)-tensor field; the metric must be used to
raise an index to obtain a (1,1)-tensor field in order to take the trace. In terms of
local coordinates
Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples:
* Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow ...
one can write, using the
Einstein notation convention, that:
:
where are the components of the Ricci tensor in the coordinate basis, and where are the
inverse metric components, i.e. the components of the
inverse of the matrix of metric components . Based upon the Ricci curvature being a sum of
sectional curvatures, it is possible to also express the scalar curvature as
:
where denotes the sectional curvature and is any
orthonormal frame at . By similar reasoning, the scalar curvature is twice the trace of the
curvature operator. Alternatively, given the coordinate-based definition of Ricci curvature in terms of the
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
, it is possible to express scalar curvature as
:
where
are the
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
of the metric, and
is the partial derivative of
in the σ-coordinate direction.
The above definitions are equally valid for a
pseudo-Riemannian metric
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 the ...
. The special case of
Lorentzian metrics is significant in the mathematical theory of
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 the scalar curvature and Ricci curvature are the fundamental terms in the
Einstein field equation.
However, unlike the
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
or the Ricci tensor, the scalar curvature cannot be defined for an arbitrary
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
, for the reason that the trace of a (0,2)-tensor field is ill-defined. However, there are other generalizations of scalar curvature, including in
Finsler geometry.
Traditional notation
In the context of
tensor index notation
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 be c ...
, it is common to use the letter to represent three different things:
# the Riemann curvature tensor: or
# the Ricci tensor:
# the scalar curvature:
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. Other notations used for scalar curvature include , , , , or , and .
Those not using an index notation usually reserve ''R'' for the full Riemann curvature tensor. Alternatively, in a coordinate-free notation one may use ''Riem'' for the Riemann tensor, ''Ric'' for the Ricci tensor and ''R'' for the scalar curvature.
Some authors instead define Ricci curvature and scalar curvature with a normalization factor, so that
:
The purpose of such a choice is that the Ricci and scalar curvatures become ''average values'' (rather than sums) of sectional curvatures.
Basic properties
It is a fundamental fact that the scalar curvature is invariant under
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
. To be precise, if is a diffeomorphism from a space to a space , the latter being equipped with a (pseudo-)Riemannian metric , then the scalar curvature of the
pullback metric on equals the composition of the scalar curvature of with the map . This amounts to the assertion that the scalar curvature is geometrically well-defined, independent of any choice of coordinate chart or local frame. More generally, as may be phrased in the language of
homotheties, the effect of scaling the metric by a constant factor is to scale the scalar curvature by the inverse factor .
Furthermore, the scalar curvature is (up to an arbitrary choice of normalization factor) the only coordinate-independent function of the metric which, as evaluated at the center of a
normal coordinate chart, is a polynomial in derivatives of the metric and has the above scaling property. This is one formulation of the
Vermeil theorem.
Bianchi identity
As a direct consequence of the
Bianchi identities, any (pseudo-)Riemannian metric has the property that
:
This identity is called the ''contracted Bianchi identity''. It has, as an almost immediate consequence, the
Schur lemma stating that if the Ricci tensor is pointwise a multiple of the metric, then the metric must be
Einstein (unless the dimension is two). Moreover, this says that (except in two dimensions) a metric is Einstein if and only if the Ricci tensor and scalar curvature are related by
:
where denotes the dimension. The contracted Bianchi identity is also fundamental in the mathematics of general relativity, since it identifies the
Einstein tensor as a fundamental quantity.
Ricci decomposition
Given a (pseudo-)Riemannian metric on a space of dimension , the ''scalar curvature part'' of the
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
is the (0,4)-tensor field
:
(This follows the convention that .) This tensor is significant as part of the
Ricci decomposition; it is orthogonal to the difference between the Riemann tensor and itself. The other two parts of the Ricci decomposition correspond to the components of the Ricci curvature which do ''not'' contribute to scalar curvature, and to the
Weyl tensor, which is the part of the Riemann tensor which does not contribute to the Ricci curvature. Put differently, the above tensor field is the only part of the Riemann curvature tensor which contributes to the scalar curvature; the other parts are orthogonal to it and make no such contribution. There is also a Ricci decomposition for the curvature of a
Kähler metric Kähler may refer to:
;People
*Alexander Kähler (born 1960), German television journalist
*Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and arc ...
.
Basic formulas
The scalar curvature of a
conformally changed metric can be computed:
:
using the convention for the
Laplace–Beltrami operator. Alternatively,
:
Under an infinitesimal change of the underlying metric, one has
:
This shows in particular that the
principal symbol of the
differential operator which sends a metric to its scalar curvature is given by
:
Furthermore the adjoint of the linearized scalar curvature operator is
:
and it is an overdetermined
elliptic operator in the case of a Riemannian metric. It is a straightforward consequence of the first variation formulas that, to first order, a Ricci-flat Riemannian metric on a
closed manifold cannot be deformed so as to have either positive or negative scalar curvature. Also to first order, an Einstein metric on a closed manifold cannot be deformed under a volume normalization so as to increase or decrease scalar curvature.
Relation between volume and Riemannian scalar curvature
When the scalar curvature is positive at a point, the volume of a small geodesic ball about the point has smaller volume than a ball of the same radius in Euclidean space. On the other hand, when the scalar curvature is negative at a point, the volume of a small ball is larger than it would be in Euclidean space.
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
. Namely, the ratio of the ''n''-dimensional volume of a ball of radius ε in the manifold to that of a corresponding ball in Euclidean space is given, for small ε, by
:
Thus, the second derivative of this ratio, evaluated at radius ''ε'' = 0, is exactly minus the scalar curvature divided by 3(''n'' + 2).
Boundaries of these balls are (''n'' − 1)-dimensional
spheres of radius
; their hypersurface measures ("areas") satisfy the following equation:
:
These expansions generalize certain
characterizations of Gaussian curvature from dimension two to higher dimensions.
Special cases
Surfaces
In two dimensions, scalar curvature is exactly twice the Gaussian curvature. For an embedded surface in Euclidean space R
3, this means that
:
where
are the
principal radii of the surface. For example, the scalar curvature of the 2-sphere of radius ''r'' is equal to 2/''r''
2.
The 2-dimensional Riemann curvature tensor has only one independent component, and it can be expressed
in terms of the scalar curvature and metric area form. Namely, in any coordinate system, one has
:
Space forms
A
space form is by definition a Riemannian manifold with constant sectional curvature. Space forms are locally isometric to one of the following types:
The scalar curvature is also constant when given a
Kähler metric Kähler may refer to:
;People
*Alexander Kähler (born 1960), German television journalist
*Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and arc ...
of constant
holomorphic sectional curvature.
Products
The scalar curvature of a
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
''M'' × ''N'' of Riemannian manifolds is the sum of the scalar curvatures of ''M'' and ''N''. For example, for any
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
closed manifold ''M'', ''M'' × ''S''
2 has a metric of positive scalar curvature, simply by taking the 2-sphere to be small compared to ''M'' (so that its curvature is large). This example might suggest that scalar curvature has little relation to the global geometry of a manifold. In fact, it does have some global significance, as discussed
below.
In both mathematics and general relativity,
warped product metrics are an important source of examples. For example, the general
Robertson–Walker spacetime, important to
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
, is the Lorentzian metric
:
on , where is a
constant-curvature Riemannian metric on a three-dimensional manifold . The scalar curvature of the Robertson–Walker metric is given by
:
where is the constant curvature of .
Scalar-flat spaces
It is automatic that any
Ricci-flat manifold has zero scalar curvature; the best-known spaces in this class are the
Calabi–Yau manifolds. In the pseudo-Riemannian context, this also includes the
Schwarzschild spacetime and
Kerr spacetime.
There are metrics with zero scalar curvature but nonvanishing Ricci curvature. For example, there is a complete Riemannian metric on the
tautological line bundle over
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properties Construction
A ...
, constructed as a
warped product metric, which has zero scalar curvature but nonzero Ricci curvature. This may also be viewed as a rotationally symmetric Riemannian metric of zero scalar curvature on the cylinder .
Yamabe problem
The ''Yamabe problem'' was resolved in 1984 by the combination of results found by
Hidehiko Yamabe
was a Japanese mathematician. Above all, he is famous for discovering that every conformal class on a smooth compact manifold is represented by a Riemannian metric of constant scalar curvature. Other notable contributions include his definitive ...
,
Neil Trudinger,
Thierry Aubin
Thierry Aubin (6 May 1942 – 21 March 2009) was a French mathematician who worked at the Centre de Mathématiques de Jussieu, and was a leading expert on Riemannian geometry
and non-linear partial differential equations. His fundamental contrib ...
, and
Richard Schoen. They proved that every smooth Riemannian metric on a
closed manifold can be multiplied by some smooth positive function to obtain a metric with constant scalar curvature. In other words, every Riemannian metric on a closed manifold is
conformal to one with constant scalar curvature.
Riemannian metrics of positive scalar curvature
For a closed Riemannian 2-manifold ''M'', the scalar curvature has a clear relation to the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of ''M'', expressed by the
Gauss–Bonnet theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology.
In the simplest application, the case of a t ...
: the total scalar curvature of ''M'' is equal to 4 times the
Euler characteristic of ''M''. For example, the only closed surfaces with metrics of positive scalar curvature are those with positive Euler characteristic: the sphere ''S''
2 and
RP2. Also, those two surfaces have no metrics with scalar curvature ≤ 0.
Nonexistence results
In the 1960s,
André Lichnerowicz
André Lichnerowicz (January 21, 1915, Bourbon-l'Archambault – December 11, 1998, Paris) was a noted French differential geometer and mathematical physicist of Polish descent. He is considered the founder of modern Poisson geometry.
Biograp ...
found that on a
spin manifold, the difference between the square of the
Dirac operator and the
tensor Laplacian (as defined on spinor fields) is given exactly by one-quarter of the scalar curvature. This is a fundamental example of a
Weitzenböck formula. As a consequence, if a Riemannian metric on a closed manifold has positive scalar curvature, then there can exist no
harmonic spinor
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally ...
s. It is then a consequence of the
Atiyah–Singer index theorem that, for any closed spin manifold with dimension divisible by four and of positive scalar curvature, the
 genus
In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the ...
must vanish. This is a purely topological obstruction to the existence of Riemannian metrics with positive scalar curvature.
Lichnerowicz's argument using the
Dirac operator can be "twisted" by an auxiliary
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
, with the effect of only introducing one extra term into the Lichnerowicz formula. Then, following the same analysis as above except using the families version of the index theorem and a refined version of the  genus known as the ''α-genus'',
Nigel Hitchin proved that in certain dimensions there are
exotic spheres which do not have any Riemannian metrics of positive scalar curvature. Gromov and Lawson later extensively employed these variants of Lichnerowicz's work. One of their resulting theorems introduces the homotopy-theoretic notion of ''enlargeability'' and says that an enlargeable spin manifold cannot have a Riemannian metric of positive scalar curvature. As a corollary, a closed manifold with a Riemannian metric of nonpositive curvature, such as a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
, has no metric with positive scalar curvature. Gromov and Lawson's various results on nonexistence of Riemannian metrics with positive scalar curvature support a conjecture on the vanishing of a wide variety of topological invariants of any closed spin manifold with positive scalar curvature. This (in a precise formulation) in turn would be a special case of the
strong Novikov conjecture for the
fundamental group, which deals with the
K-theory of C*-algebras. This in turn is a special case of the
Baum–Connes conjecture for the fundamental group.
In the special case of four-dimensional manifolds, the
Seiberg–Witten equations have been usefully applied to the study of scalar curvature. Similarly to Lichnerowicz's analysis, the key is an application of the
maximum principle to prove that solutions to the Seiberg–Witten equations must be trivial when scalar curvature is positive. Also in analogy to Lichnerowicz's work, index theorems can guarantee the existence of nontrivial solutions of the equations. Such analysis provides new criteria for nonexistence of metrics of positive scalar curvature.
Claude LeBrun
Claude R. LeBrun (born 1956) is an American mathematician who holds the position of SUNY Distinguished Professor of Mathematics at Stony Brook University. Much of his research concerns the Riemannian geometry of 4-manifolds, or related topics in ...
pursued such ideas in a number of papers.
Existence results
By contrast to the above nonexistence results, Lawson and Yau constructed Riemannian metrics of positive scalar curvature from a wide class of nonabelian effective group actions.
Later, Schoen–Yau and Gromov–Lawson (using different techniques) proved the fundamental result that existence of Riemannian metrics of positive scalar curvature is preserved by
topological surgery in codimension at least three, and in particular is preserved by the
connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
. This establishes the existence of such metrics on a wide variety of manifolds. For example, it immediately shows that the connected sum of an arbitrary number of copies of
spherical space forms and generalized cylinders has a Riemannian metric of positive scalar curvature.
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 ...
's construction of
Ricci flow with surgery has, as an immediate corollary, the converse in the three-dimensional case: a closed
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
3-manifold with a Riemannian metric of positive scalar curvature must be such a connected sum.
Based upon the surgery allowed by the Gromov–Lawson and Schoen–Yau construction, Gromov and Lawson observed that the
h-cobordism theorem
In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps
: M ...
and analysis of the
cobordism ring can be directly applied. They proved that, in dimension greater than four, any non-spin
simply connected closed manifold has a Riemannian metric of positive scalar curvature. Stephan Stolz completed the existence theory for simply-connected closed manifolds in dimension greater than four, showing that as long as the α-genus is zero, then there is a Riemannian metric of positive scalar curvature.
According to these results, for closed manifolds, the existence of Riemannian metrics of positive scalar curvature is completely settled in the three-dimensional case and in the case of simply-connected manifolds of dimension greater than four.
Kazdan and Warner's trichotomy theorem
The sign of the scalar curvature has a weaker relation to topology in higher dimensions. Given a smooth closed manifold ''M'' of dimension at least 3,
Kazdan and Warner solved the
prescribed scalar curvature problem, describing which smooth functions on ''M'' arise as the scalar curvature of some Riemannian metric on ''M''. Namely, ''M'' must be of exactly one of the following three types:
# Every function on ''M'' is the scalar curvature of some metric on ''M''.
# A function on ''M'' is the scalar curvature of some metric on ''M'' if and only if it is either identically zero or negative somewhere.
# A function on ''M'' is the scalar curvature of some metric on ''M'' if and only if it is negative somewhere.
Thus every manifold of dimension at least 3 has a metric with negative scalar curvature, in fact of constant negative scalar curvature. Kazdan–Warner's result focuses attention on the question of which manifolds have a metric with positive scalar curvature, that being equivalent to property (1). The borderline case (2) can be described as the class of manifolds with a strongly scalar-flat metric, meaning a metric with scalar curvature zero such that ''M'' has no metric with positive scalar curvature.
Akito Futaki showed that strongly scalar-flat metrics (as defined above) are extremely special. For a simply connected Riemannian manifold ''M'' of dimension at least 5 which is strongly scalar-flat, ''M'' must be a product of Riemannian manifolds with
holonomy
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
group SU(''n'') (
Calabi–Yau manifolds), Sp(''n'') (
hyperkähler manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold (M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the Riemannian metric g and satisfy the quaternionic relations I^2 ...
s), or Spin(7).
[Petersen (2016), Corollary C.4.4.] In particular, these metrics are Ricci-flat, not just scalar-flat. Conversely, there are examples of manifolds with these holonomy groups, such as the
K3 surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected al ...
, which are spin and have nonzero α-invariant, hence are strongly scalar-flat.
See also
*
Basic introduction to the mathematics of curved spacetime
*
Yamabe invariant
*
Kretschmann scalar
Notes
References
*
*
*
*
*
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Further reading
*
*
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{{DEFAULTSORT:Scalar Curvature
Curvature (mathematics)
Riemannian geometry
Trace theory
de:Riemannscher Krümmungstensor#Krümmungsskalar