Pullback Riemannian metrics #
Let b be a family of bilinear forms on the tangent spaces of a manifold N, for instance a
Riemannian metric, that is a C^n section near f x₀, where f : M → N is a map that is
C^(n+1) at a point x₀. Pulling b back along the differential of f gives at each point x
the bilinear form (v, w) ↦ b_{f x}(df_x v, df_x w) on T_x M. This file proves that the family
of these forms is C^n at x₀, as a section of the bundle of bilinear forms on the tangent bundle
of M.
No injectivity of the differential is assumed: the statement is only about smoothness. When b is
positive definite (for instance a Riemannian metric), the pulled-back forms are positive definite
exactly where df is injective. This is the regularity
input for Riemannian metrics that are defined by pulling back along local diffeomorphisms, such as
the metric that a manifold with a geometric structure receives through its charts.
Main results #
Bundle.trivializationAt_bilinearForm_apply: a bilinear form onT_x M, read in the bundle chart atx₀, is pulled back along the tangent coordinate change fromx₀tox.Bundle.contMDiffAt_trivializationAt_bilinearForm_pullback: the pullback of aC^nfamily of bilinear forms along linear maps that areC^nin tangent coordinates, both depending on a parameter, isC^nwhen read in a bundle chart. This gives the joint regularity of pullbacks along families of maps.Bundle.contMDiffAt_bilinearForm_pullback_of_inTangentCoordinates: the pullback of aC^nfamily of bilinear forms along a family of linear maps that isC^nin tangent coordinates is aC^nsection.Bundle.contMDiffAt_bilinearForm_pullback: the pullback of a family of bilinear forms that isC^natf x₀, along a mapfthat isC^(n+1)atx₀, is aC^nsection atx₀.Bundle.ContMDiffRiemannianMetric.contMDiffAt_pullback: the special case of aC^nRiemannian metric.Bundle.ContMDiffRiemannianMetric.pullback: theC^nRiemannian metric obtained from an immersion by pulling back the target metric.Bundle.ContMDiffRiemannianMetric.pullback_idandBundle.ContMDiffRiemannianMetric.pullback_comp: the functoriality laws for pullback metrics.Bundle.ContMDiffRiemannianMetric.pullback_rescale: pulling back commutes with conformal rescaling,f^* (h g) = (h ∘ f) f^* g.Bundle.ContMDiffRiemannianMetric.conformalClass_pullback: along a map with aC^nleft inverse, such as a diffeomorphism, the conformal class off^* gis the pullback of the conformal class ofg.
References #
- J. M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer GTM 176 (2018), Chapter 2 (pullback metrics).
Bilinear forms on tangent spaces in bundle coordinates #
A bilinear form on T_x M, read in the trivialization of the bundle of bilinear forms at
x₀, is pulled back along the tangent coordinate change from the chart at x₀ to the chart at
x.
Read in the trivialization at x itself, a bilinear form on T_x M is unchanged.
A bilinear form on T_x M, read in the trivialization at x₀, as a continuous linear
function of the form.
Pullbacks of bilinear forms #
Pull back a family b p of bilinear forms on the tangent spaces T_{f p} N, depending on a
parameter p and C^n along f at p₀, along a family of linear maps
L p : T_{g p} M → T_{f p} N that is C^n at p₀ in tangent coordinates. Read in the
trivialization of the bundle of bilinear forms on T M at g p₀, the resulting family of
bilinear forms (v, w) ↦ b p (L p v, L p w) on T_{g p} M is C^n at p₀.
For g = id this says that the pulled-back forms are a C^n section
(Bundle.contMDiffAt_bilinearForm_pullback_of_inTangentCoordinates). Other choices of g, such
as a projection M × P → M, give the joint regularity of pullbacks along families of maps.
Pull back a family b x of bilinear forms on the tangent spaces T_{f x} N, a C^n section
along f at x₀, along a family of linear maps L x : T_x M → T_{f x} N that is C^n at x₀ in
tangent coordinates. The result, the family of bilinear forms (v, w) ↦ b x (L x v, L x w), is a
C^n section at x₀.
The pullback of a family b of bilinear forms on the tangent spaces of N that is a C^n
section at f x₀, along a map f that is C^(n+1) at x₀, is a C^n section at x₀: the family
of bilinear forms (v, w) ↦ b_{f x}(df_x v, df_x w).
The pullback of a C^n Riemannian metric g along a map f that is C^(n+1) at x₀, the
family of bilinear forms (v, w) ↦ g_{f x}(df_x v, df_x w), is a C^n section at x₀.
Pull back a smooth Riemannian metric along a smooth immersion. At x, the inner product is
the target metric at f x evaluated on the two images under the manifold derivative of f.
Injectivity of the derivative makes this form positive definite.
Instances For
The pullback metric is obtained by applying the target metric to the two differentials.
Pulling back along equal maps gives equal metrics, whatever the regularity and injectivity proofs supplied.
Pulling a Riemannian metric back along the identity leaves it unchanged.
Pulling a metric back along a composite is the same as pulling it back successively.
Pulling back commutes with conformal rescaling: the pullback of h g is (h ∘ f) times the
pullback of g.
The pullback of a metric conformal to g is conformal to the pullback of g.
Along a map f with a C^n left inverse, for instance a C^(n+1) diffeomorphism, the
conformal class of the pullback of g is the pullback of the conformal class of g.