Curvature under Riemannian isometries #
A local Riemannian isometry f : M → N, and in particular a smooth Riemannian isometry, carries
the Riemann curvature tensor of M to that of N:
df_x (R(u, v) w) = R(df_x u, df_x v) (df_x w)
for all tangent vectors u, v, w at x. Consequently it preserves every curvature quantity built
from the curvature tensor and the metric: the sectional curvature of each tangent two-plane, the
Ricci tensor and the scalar curvature. Since f need not be injective or surjective, this applies
to the inclusion of an open submanifold with the restricted metric, so these curvatures are local
invariants of the metric.
These invariants are what constrain an isometry of a homogeneous space that is not a space form.
For example, on S² × ℝ the tangent planes of the sphere factor are exactly the planes of
sectional curvature 1, and the line field tangent to the ℝ factor is the kernel of the Ricci
tensor, so the differential of an isometry preserves the product splitting of each tangent space.
Combined with the determination of isometries by their first-order data, this is the input for
identifying the full isometry groups of such model spaces.
The curvature identity follows from the naturality of the Levi-Civita connection
(TauCeti.IsLocalRiemannianIsometry.mfderiv_leviCivitaConnection_mpullback) and of the Lie
bracket (VectorField.mpullback_mlieBracket): pulling back three smooth vector fields along f
pulls back each of the three terms in R(X, Y) Z = ∇_X ∇_Y Z - ∇_Y ∇_X Z - ∇_[X, Y] Z.
Main results #
In the namespace TauCeti.IsLocalRiemannianIsometry:
mfderiv_curvatureOperator_mpullback: the curvature operator of pulled-back smooth vector fields is the pullback of the curvature operator;curvatureOperator_mpullbackis its vector-field form.mfderiv_curvatureTensor: naturality of the Riemann curvature tensor.inner_curvatureTensor_mfderivandsectionalCurvature_mfderiv: preservation of the curvature tensor paired with the metric, and of sectional curvature.ricciTensor_mfderivandscalarCurvature_map: preservation of the Ricci tensor and of the scalar curvature.
These apply to a Riemannian isometry Φ, which is a local Riemannian isometry
(Φ.isLocalRiemannianIsometry). In the namespace TauCeti.RiemannianIsometry,
hasSectionalCurvatureAt_map_iff and hasConstantSectionalCurvature_iff use surjectivity:
isometries preserve pointwise and constant sectional curvature.
References #
- J. M. Lee, Introduction to Riemannian Manifolds, 2nd ed., GTM 176, 2018, Proposition 7.6 (naturality of the curvature tensor).
- M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992, Ch. 4, §3.
Naturality of the curvature operator under local Riemannian isometries. For smooth
vector fields X, Y, Z on N, the curvature operator of M applied to their pullbacks along
f is the pullback of the curvature operator of N:
df_x (R(f^* X, f^* Y) (f^* Z) x) = R(X, Y) Z (f x).
The vector-field form of the naturality of the curvature operator: for smooth vector fields
X, Y, Z on N, R(f^* X, f^* Y) (f^* Z) = f^* (R(X, Y) Z).
Naturality of the Riemann curvature tensor under local Riemannian isometries:
df_x (R(u, v) w) = R(df_x u, df_x v) (df_x w).
A local Riemannian isometry preserves the curvature tensor paired with the metric:
⟪R(df_x u, df_x v) (df_x w), df_x z⟫ = ⟪R(u, v) w, z⟫.
A local Riemannian isometry preserves sectional curvature: the plane spanned by df_x u and
df_x v has the same sectional curvature as the plane spanned by u and v.
A local Riemannian isometry preserves the Ricci tensor:
Ric(df_x u, df_x v) = Ric(u, v).
A local Riemannian isometry preserves scalar curvature.
A Riemannian isometry Φ preserves pointwise sectional curvature: N has sectional
curvature k at Φ x exactly when M has sectional curvature k at x.
Riemannian isometric manifolds have the same constant sectional curvatures.