Positive scalar curvature #
A smooth Riemannian metric g on the tangent bundle has positive scalar curvature when its
scalar curvature Bundle.ContMDiffRiemannianMetric.scalarCurvature, that of the Levi-Civita
connection of g, is positive at every point. Since scalar curvature is a smooth function of
the point, on a compact manifold such a metric has scalar curvature bounded below by a positive
constant. Scalar curvature is invariant under diffeomorphisms, so pulling a metric back along a
diffeomorphism preserves and reflects positive scalar curvature. Likewise a metric descended along
a surjective local diffeomorphism, such as the metric induced on the orbit space of a free,
properly discontinuous action by isometries, has positive scalar curvature exactly when the
original metric does. Scalar curvature is also local, so the restriction of a metric of positive
scalar curvature to an open submanifold has positive scalar curvature. No assumption on the
boundary is made. The disjoint union of two metrics has the scalar curvature of each metric on its
summand, so it has positive scalar curvature exactly when both metrics do.
Since Scal(c g) = c⁻¹ Scal(g) for a constant c > 0, rescaling by a positive constant preserves
and reflects positive scalar curvature. The scalar curvature of a product metric g₁ × g₂ is the
sum of the scalar curvatures of the factors, so g₁ × g₂ has positive scalar curvature when g₁
does and g₂ has nonnegative scalar curvature, as for a flat factor. More importantly,
shrinking a compact factor of positive scalar curvature makes its contribution c⁻¹ Scal(g₂)
dominate any scalar curvature of the other factor that is bounded below: if N is compact, g₂
has positive scalar curvature and the scalar curvature of g₁ is bounded below, then g₁ × c g₂
has positive scalar curvature for every sufficiently small c > 0.
On a compact manifold the scalar curvature of every smooth metric is bounded below.
Main definitions and results #
Bundle.ContMDiffRiemannianMetric.HasPositiveScalarCurvature: positivity at every point.Bundle.ContMDiffRiemannianMetric.HasPositiveScalarCurvature.exists_pos_le: on a compact manifold, positive scalar curvature has a uniform positive lower bound.Bundle.ContMDiffRiemannianMetric.hasPositiveScalarCurvature_iff_exists_pos_le: on a compact manifold, positive scalar curvature is equivalent to a uniform positive lower bound.Bundle.ContMDiffRiemannianMetric.hasPositiveScalarCurvature_pullback_iff: the pullback of a metric along a diffeomorphism has positive scalar curvature exactly when the metric does.Bundle.ContMDiffRiemannianMetric.HasPositiveScalarCurvature.pullback_of_isInvertible: the pullback of a metric of positive scalar curvature along a smooth map with invertible differential has positive scalar curvature.Bundle.ContMDiffRiemannianMetric.hasPositiveScalarCurvature_descend_iffandBundle.ContMDiffRiemannianMetric.hasPositiveScalarCurvature_quotient_iff: a metric descended along a surjective local diffeomorphism, in particular to the orbit space of a free, properly discontinuous action by isometries, has positive scalar curvature exactly when the metric does.Bundle.ContMDiffRiemannianMetric.HasPositiveScalarCurvature.restrictOpenTangentSpace: the restriction of a metric of positive scalar curvature to an open submanifold has positive scalar curvature.Bundle.ContMDiffRiemannianMetric.HasPositiveScalarCurvature.restrictIcc: the restriction of a metric of positive scalar curvature onM × ℝtoM × [a, b]has positive scalar curvature.Bundle.ContMDiffRiemannianMetric.hasPositiveScalarCurvature_sumTangentSpace_iff: the disjoint union of two metrics has positive scalar curvature exactly when both metrics do.Bundle.ContMDiffRiemannianMetric.hasPositiveScalarCurvature_rescale_const_iff: rescaling a metric by a positive constant preserves and reflects positive scalar curvature.Bundle.ContMDiffRiemannianMetric.HasPositiveScalarCurvature.prodTangentSpace_of_nonneg: the product of a metric of positive scalar curvature with one of nonnegative scalar curvature has positive scalar curvature, andBundle.ContMDiffRiemannianMetric.HasPositiveScalarCurvature.prodTangentSpace_of_nonneg_leftthe same with the factors swapped.Bundle.ContMDiffRiemannianMetric.exists_pos_hasPositiveScalarCurvature_prodTangentSpace_rescale: a compact factor of positive scalar curvature, scaled down, dominates a factor whose scalar curvature is bounded below; more generallyexists_pos_hasPositiveScalarCurvature_prodTangentSpace_rescale_of_forall_ledoes so for a factor whose scalar curvature has a positive lower bound, andexists_pos_forall_hasPositiveScalarCurvature_prodTangentSpace_rescale_of_forall_ledoes so uniformly for a family of metrics with a common lower bound.
References #
- M. Gromov, H. B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. 111 (1980), 423–434.
- J. M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer GTM 176 (2018), Chapter 7, for the scalar-curvature convention.
A smooth Riemannian metric has positive scalar curvature if its scalar curvature is positive at every point.
Instances For
A metric has positive scalar curvature exactly when its scalar curvature is positive at every point.
On a compact manifold, a metric of positive scalar curvature has scalar curvature bounded below by a positive constant.
On a compact manifold, a metric has positive scalar curvature exactly when its scalar curvature has a uniform positive lower bound.
The pullback of a metric along a diffeomorphism has positive scalar curvature exactly when the metric does.
Pulling a metric of positive scalar curvature back along a diffeomorphism gives a metric of positive scalar curvature.
Pulling a metric of positive scalar curvature back along a smooth map whose differential is invertible at every point gives a metric of positive scalar curvature.
A metric descended along a surjective local diffeomorphism has positive scalar curvature exactly when the original metric does.
The metric on the orbit space of a free, properly discontinuous smooth action by isometries has positive scalar curvature exactly when the original metric does.
The restriction of a metric of positive scalar curvature to an open submanifold has positive scalar curvature.
The restriction to M × [a, b] of a metric of positive scalar curvature on M × ℝ has
positive scalar curvature.
The disjoint union of two metrics has positive scalar curvature exactly when both metrics do.
The disjoint union of two metrics of positive scalar curvature has positive scalar curvature.
Rescaling a metric by a positive constant preserves and reflects positive scalar curvature.
Rescaling a metric of positive scalar curvature by a positive constant gives a metric of positive scalar curvature.
The product of a metric of positive scalar curvature with a metric of nonnegative scalar curvature has positive scalar curvature.
The product of a metric of nonnegative scalar curvature with a metric of positive scalar curvature has positive scalar curvature.
Shrinking a factor of uniformly positive scalar curvature, uniformly in a family. Let
g₂ be a metric whose scalar curvature is at least δ > 0, and let f i be metrics whose
scalar curvatures have a common lower bound m. Then there is ε > 0 such that f i × c g₂ has
positive scalar curvature for every i and every 0 < c < ε.
Shrinking a factor of uniformly positive scalar curvature. Let g₂ be a metric whose
scalar curvature is at least δ > 0, and let g be a metric whose scalar curvature is bounded
below. Then there is ε > 0 such that g × c g₂ has positive scalar curvature for every
0 < c < ε.
Shrinking a compact factor of positive scalar curvature. Let g₂ be a metric of positive
scalar curvature on a compact manifold N, and let g be a metric whose scalar curvature is
bounded below. Then there is ε > 0 such that g × c g₂ has positive scalar curvature for every
0 < c < ε.