In
Riemannian geometry, the sectional curvature is one of the ways to describe the
curvature of Riemannian manifolds
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigoro ...
. The sectional curvature ''K''(σ
''p'') depends on a two-dimensional linear subspace σ
''p'' of the
tangent space at a point ''p'' of the manifold. It can be defined geometrically as the
Gaussian curvature of the
surface which has the plane σ
''p'' as a tangent plane at ''p'', obtained from
geodesics which start at ''p'' in the directions of σ
''p'' (in other words, the image of σ
''p'' under the
exponential map at ''p''). The sectional curvature is a real-valued function on the 2-
Grassmannian bundle over the manifold.
The sectional curvature determines the
curvature tensor completely.
Definition
Given a
Riemannian manifold and two
linearly independent tangent vectors at the same point, ''u'' and ''v'', we can define
:
Here ''R'' is the
Riemann curvature tensor, defined here by the convention
Some sources use the opposite convention
in which case ''K(u,v)'' must be defined with
in the numerator instead of
Note that the linear independence of ''u'' and ''v'' forces the denominator in the above expression to be nonzero, so that ''K(u,v)'' is well-defined. In particular, if ''u'' and ''v'' are
orthonormal, then the definition takes on the simple form
:
It is straightforward to check that if
are linearly independent and span the same two-dimensional linear subspace of the
tangent space as
, then
So one may consider the sectional curvature as a real-valued function whose input is a two-dimensional linear subspace of a tangent space.
Alternative definitions
Alternatively, the sectional curvature can be characterized by the circumference of small circles. Let
be a two-dimensional plane in
. Let
for sufficiently small
denote the image under the exponential map at
of the unit circle in
, and let
denote the length of
. Then it can be proven that
:
as
, for some number
. This number
at
is the sectional curvature of
at
.
Manifolds with constant sectional curvature
One says that a Riemannian manifold has "constant curvature
" if
for all two-dimensional linear subspaces
and for all
The
Schur lemma states that if ''(M,g)'' is a connected Riemannian manifold with dimension at least three, and if there is a function
such that
for all two-dimensional linear subspaces
and for all
then ''f'' must be constant and hence ''(M,g)'' has constant curvature.
A Riemannian manifold with constant sectional curvature is called a
space form. If
denotes the constant value of the sectional curvature, then the curvature tensor can be written as
:
for any
Since any Riemannian metric is parallel with respect to its Levi-Civita connection, this shows that the Riemann tensor of any constant-curvature space is also parallel. The Ricci tensor is then given by
and the scalar curvature is
In particular, any constant-curvature space is Einstein and has constant scalar curvature.
The model examples
Given a positive number
define
*
to be the standard Riemannian structure
*
to be the sphere
with
given by the pullback of the standard Riemannian structure on
by the inclusion map
*
to be the ball
with
In the usual terminology, these Riemannian manifolds are referred to as
Euclidean space, the
n-sphere, and
hyperbolic space. Here, the point is that each is a complete connected smooth Riemannian manifold with constant curvature. To be precise, the Riemannian metric
has constant curvature 0, the Riemannian metric
has constant curvature
and the Riemannian metric
has constant curvature
Furthermore, these are the 'universal' examples in the sense that if
is a smooth, connected, and simply-connected complete Riemannian manifold with constant curvature, then it is isometric to one of the above examples; the particular example is dictated by the value of the constant curvature of
according to the constant curvatures of the above examples.
If
is a smooth and connected complete Riemannian manifold with constant curvature, but is ''not'' assumed to be simply-connected, then consider the universal covering space
with the pullback Riemannian metric
Since
is, by topological principles, a covering map, the Riemannian manifold
is locally isometric to
, and so it is a smooth, connected, and simply-connected complete Riemannian manifold with the same constant curvature as
It must then be isometric one of the above model examples. Note that the deck transformations of the universal cover are
isometries relative to the metric
The study of Riemannian manifolds with constant negative curvature, called
hyperbolic geometry, is particularly noteworthy as it exhibits many noteworthy phenomena.
Scaling
Let
be a smooth manifold, and let
be a positive number. Consider the Riemannian manifold
The curvature tensor, as a multilinear map
is unchanged by this modification. Let
be linearly independent vectors in
. Then
:
So multiplication of the metric by
multiplies all of the sectional curvatures by
Toponogov's theorem
Toponogov's theorem
In the mathematical field of Riemannian geometry, Toponogov's theorem (named after Victor Andreevich Toponogov) is a triangle comparison theorem.
It is one of a family of comparison theorems that quantify the assertion that a pair of geodesics em ...
affords a characterization of sectional curvature in terms of how "fat"
geodesic triangles appear when compared to their Euclidean counterparts. The basic intuition is that, if a space is positively curved, then the edge of a triangle opposite some given vertex will tend to bend away from that vertex, whereas if a space is negatively curved, then the opposite edge of the triangle will tend to bend towards the vertex.
More precisely, let ''M'' be a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
Riemannian manifold, and let ''xyz'' be a geodesic triangle in ''M'' (a triangle each of whose sides is a length-minimizing geodesic). Finally, let ''m'' be the midpoint of the geodesic ''xy''. If ''M'' has non-negative curvature, then for all sufficiently small triangles
:
where ''d'' is the
distance function on ''M''. The case of equality holds precisely when the curvature of ''M'' vanishes, and the right-hand side represents the distance from a vertex to the opposite side of a geodesic triangle in Euclidean space having the same side-lengths as the triangle ''xyz''. This makes precise the sense in which triangles are "fatter" in positively curved spaces. In non-positively curved spaces, the inequality goes the other way:
:
If tighter bounds on the sectional curvature are known, then this property generalizes to give a
comparison theorem In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry.
Differential e ...
between geodesic triangles in ''M'' and those in a suitable simply connected space form; see
Toponogov's theorem
In the mathematical field of Riemannian geometry, Toponogov's theorem (named after Victor Andreevich Toponogov) is a triangle comparison theorem.
It is one of a family of comparison theorems that quantify the assertion that a pair of geodesics em ...
. Simple consequences of the version stated here are:
* A complete Riemannian manifold has non-negative sectional curvature if and only if the function
is 1-
concave for all points ''p''.
* A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function
is 1-
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polyto ...
.
Manifolds with non-positive sectional curvature
In 1928,
Élie Cartan proved the
Cartan–Hadamard theorem
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is dif ...
: if ''M'' is a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
manifold with non-positive sectional curvature, then its
universal cover is
diffeomorphic to a
Euclidean space. In particular, it is
aspherical: the
homotopy groups for ''i'' ≥ 2 are trivial. Therefore, the topological structure of a complete non-positively curved manifold is determined by its
fundamental group.
Preissman's theorem restricts the fundamental group of negatively curved compact manifolds. The
Cartan–Hadamard conjecture states that the classical
isoperimetric inequality should hold in all simply connected spaces of non-positive curvature, which are called
Cartan-Hadamard manifolds.
Manifolds with positive sectional curvature
Little is known about the structure of positively curved manifolds. The
soul theorem (; ) implies that a complete non-compact non-negatively curved manifold is diffeomorphic to a normal bundle over a compact non-negatively curved manifold. As for compact positively curved manifolds, there are two classical results:
* It follows from the
Myers theorem
Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following:
In the special case o ...
that the fundamental group of such a manifold is finite.
* It follows from the
Synge theorem that the fundamental group of such a manifold in even dimensions is 0, if orientable and
otherwise. In odd dimensions a positively curved manifold is always orientable.
Moreover, there are relatively few examples of compact positively curved manifolds, leaving a lot of conjectures (e.g., the
Hopf conjecture on whether there is a metric of positive sectional curvature on
). The most typical way of constructing new examples is the following corollary from the O'Neill curvature formulas: if
is a Riemannian manifold admitting a free isometric action of a Lie group G, and M has positive sectional curvature on all 2-planes orthogonal to the orbits of G, then the manifold
with the quotient metric has positive sectional curvature. This fact allows one to construct the classical positively curved spaces, being spheres and projective spaces, as well as these examples :
* The Berger spaces
and
.
* The Wallach spaces (or the homogeneous flag manifolds):
,
and
.
* The Aloff–Wallach spaces
.
* The Eschenburg spaces
* The Bazaikin spaces
, where
.
Manifolds with non-negative sectional curvature
Cheeger and Gromoll proved their soul theorem which states that any non-negatively curved complete non-compact manifold
has a totally convex compact submanifold
such that
is diffeomorphic to the normal bundle of
. Such an
is called the soul of
. In particular, this theorem implies that
is homotopic to its soul
which has the dimension less than
.
See also
*
Riemann curvature tensor
*
Curvature of Riemannian manifolds
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigoro ...
*
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canon ...
*
Holomorphic sectional curvature
References
*
*.
*
*.
*
*
*
*
{{curvature
Curvature (mathematics)
Riemannian geometry
Riemannian manifolds