The shape operator of a map into a real inner product space #
Let i : V → E be a map from a finite-dimensional real normed space into a real inner product
space whose derivative L = Di(x) is injective, so that g(v, w) = ⟪L v, L w⟫ is a metric on
V, the metric induced by i. For a vector ν : E, the shape operator S_ν is the
endomorphism of V determined by
g(S_ν v, w) = ⟪ν, B(v, w)⟫,
where B is the second fundamental form TauCeti.secondFundamentalForm ℝ i x. It depends
linearly on ν, only through the normal component of ν, and it is g-self-adjoint for a C²
map.
The shape operator controls the squared-distance function ρ_a(y) = ‖i y - a‖². At a critical
point x, where a - i x is normal, the Hessian of ρ_a is 2 g((1 - S_{a - i x}) v, w), so its
null space is the 1-eigenspace of S_{a - i x}. Along the normal line a = i x + t ν with
t ≠ 0 this is the t⁻¹-eigenspace of S_ν: such a point i x + t ν is a focal point of i
at x exactly when t⁻¹ is an eigenvalue of S_ν, with multiplicity the dimension of that
eigenspace. A zero eigenvalue of S_ν gives no focal point, and i x itself (t = 0) is never
one.
For a holomorphic immersion into a complex inner product space, with the real inner product
re ⟪x, y⟫, the complex identities of the second fundamental form give
S_ν J = -J S_ν and S_{Jν} = J S_ν,
where J is multiplication by I. Hence J identifies the κ- and -κ-eigenspaces of S_ν
and preserves its generalized 0-eigenspaces: the principal curvatures of a complex submanifold
come in pairs ±κ with equal multiplicities, and on every normal line the focal points at
distances t and -t have the same multiplicity.
Main definitions #
TauCeti.shapeOperator i x ν: the shape operatorS_νofiatx.
Main results #
TauCeti.inner_fderiv_shapeOperator_apply: the defining identityg(S_ν v, w) = ⟪ν, B(v, w)⟫.TauCeti.eq_shapeOperator_iff: the shape operator is the unique endomorphism satisfying it.ContDiffAt.inner_fderiv_shapeOperator_apply_comm: the shape operator isg-self-adjoint.TauCeti.fderiv_fderiv_norm_sq_sub_comp_apply_eq_inner_shapeOperator: the Hessian of squared distance at a critical point is2 g((1 - S_{a - i x}) v, w).TauCeti.ker_fderiv_fderiv_norm_sq_sub_comp: the null space of that Hessian is the1-eigenspace ofS_{a - i x}.TauCeti.ker_fderiv_fderiv_norm_sq_sub_add_smul_comp: at the pointi x + t νof a normal line witht ≠ 0, this is thet⁻¹-eigenspace ofS_ν.ContDiffAt.shapeOperator_apply_I_smul,ContDiffAt.shapeOperator_I_smul_apply: the identitiesS_ν J = -J S_νandS_{Jν} = J S_νfor a holomorphic immersion.ContDiffAt.finrank_genEigenspace_shapeOperator_neg: the paired principal curvatures.ContDiffAt.I_smul_mem_genEigenspace_shapeOperator_zero: the zero principal curvature has a complex eigenspace.ContDiffAt.finrank_ker_fderiv_fderiv_norm_sq_sub_sub_comp: the symmetric focal points.
References #
- J. Milnor, Morse Theory, Princeton University Press, 1963, Sections 6 and 7.
- S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. II, Chapter IX.
The shape operator S_ν of i : V → E at x in the direction ν: the endomorphism of
V with ⟪Di(x) (S_ν v), Di(x) w⟫ = ⟪ν, B(v, w)⟫ for all v and w, where B is the second
fundamental form. It is defined when Di(x) is injective, so that the induced metric
⟪Di(x) v, Di(x) w⟫ is nondegenerate, and is 0 otherwise.
Instances For
The defining identity of the shape operator. For an immersion, the induced metric pairs
S_ν v with w as ν pairs with the second fundamental form B(v, w).
The shape operator of a map whose derivative is not injective is 0 by convention.
Uniqueness of the shape operator. For an immersion, S_ν is the only endomorphism S
with ⟪Di(x) (S v), Di(x) w⟫ = ⟪ν, B(v, w)⟫ for all v and w.
The shape operator is additive in the normal direction.
The shape operator is homogeneous in the normal direction.
The shape operator depends only on the normal component of ν: tangent vectors have shape
operator 0.
The shape operator of a C² map is self-adjoint for the induced metric:
g(S_ν v, w) = g(v, S_ν w).
Squared distance and focal points #
The Hessian of squared distance at a critical point. If i x - a is normal, the Hessian
of y ↦ ‖i y - a‖² at x is 2 g((1 - S_{a - i x}) v, w), for the induced metric g.
The null space of the squared-distance Hessian. If i x - a is normal, the kernel of the
Hessian of y ↦ ‖i y - a‖² at x is the 1-eigenspace of the shape operator S_{a - i x}.
Its dimension is the multiplicity of a as a focal point of i at x.
Focal points on a normal line. For a normal vector ν and t ≠ 0, the kernel of the
Hessian at x of squared distance from i x + t • ν is the t⁻¹-eigenspace of S_ν. Thus
i x + t • ν is a focal point of i at x exactly when t⁻¹ is an eigenvalue of S_ν, with
multiplicity the dimension of that eigenspace.
The base point is not a focal point: squared distance from i x has nondegenerate Hessian
at x along an immersion.
Holomorphic immersions #
The real inner product re ⟪x, y⟫ on a complex inner product space is enabled locally; the shape
operators below are taken with respect to it.
The shape operator anticommutes with J. For a C² holomorphic map,
S_ν (I • v) = -(I • S_ν v).
Rotating the normal vector by J. For a C² holomorphic map,
S_{I • ν} v = I • S_ν v.
Paired principal curvatures. For a C² holomorphic map, multiplication by I identifies
the generalized κ- and -κ-eigenspaces of each shape operator, so they have the same
dimension. For k = 1 these are the geometric multiplicities of the principal curvatures κ
and -κ.
For a C² holomorphic map, -κ is a principal curvature in the direction ν exactly when
κ is.
The zero principal curvature. For a C² holomorphic map, the generalized
0-eigenspaces of a shape operator, in particular its kernel, are complex subspaces.
Symmetric focal points. For a holomorphic immersion and a normal vector ν, the points
i x + ν and i x - ν have the same focal multiplicity, possibly zero: one is a focal point of
i at x exactly when the other is, and the Hessians at x of squared distance from them have
null spaces of the same dimension.