Sub-Riemannian Heisenberg Groups
Rory Biggs
Geometry and Geometric Control (GGC) Research Group Department of Mathematics (Pure and Applied)
Rhodes University, Grahamstown 6140, South Africa
Debrecen University June 16, 2014
Problem statement
Context
Invariant sub-Riemannian structures on Lie groups.
Structures on nilpotent groups and Carnot groups.
Problem: structures on Heisenberg groups
Classification of sub-Riemannian (and Riemannian) structures.
Determination of isometry group for normal forms.
Computation of geodesics.
Outline
1 Introduction
2 Classification
3 Isometry group
4 Geodesics
5 Conclusion
Outline
1 Introduction
2 Classification
3 Isometry group
4 Geodesics
5 Conclusion
Invariant sub-Riemannian manifolds
Left-invariant sub-Riemannian manifold (G,D,g) Lie group G with Lie algebra g.
Left-invariant bracket generating distribution D D(g) is subspace of TgG
D(g) =T1Lg· D(1) Lie(D(1)) =g.
Left-invariant Riemannianmetric g on D
gg is a symmetric positive definite inner product on D(g) gg(T1Lg·A,T1Lg·B) =g1(A,B) for A,B∈g.
Remark
Structure (D,g) on G is fully specified by subspace D(1) of Lie algebra g inner product g on D(1).
Isometries
Isometric
(G,D,g) and (G0,D0,g0) are isometic
if there exists a diffeomorphism φ: G→G0 such that φ∗D=D0 and g=φ∗g0
Theorem (cf. [Lau99a, Lau99b, Wil82])
On simply connected nilpotent Lie groups, any isometry between
left-invariant Riemannian structures is the composition of a left translation and a Lie group isomorphism.
Theorem (cf. [Ham90, Kis03], see also [CaLeD14])
On Carnot groups, any isometry between the associated left-invariant sub-Riemannian structures is the composition of a left translation and a Lie group isomorphism.
Heisenberg group
Hn :
1 x1 x2 · · · xn z
0 1 0 0 y1
0 0 1 0 y2
... . .. ...
0 · · · 1 yn
0 · · · 0 1
=m(z,x,y), z,xi,yi ∈R
Simply connected (2n+ 1)-dimensional nilpotent Lie group.
Has one-dimensional centre: {m(z,0,0) : z ∈R}.
Two-step Carnot group.
Heisenberg Lie algebra
hn :
0 x1 x2 · · · xn z
0 0 0 0 y1
0 0 0 0 y2
... . .. ...
0 · · · 0 yn
0 · · · 0 0
=zZ +
n
X
i=1
(xiXi+yiYi), z,xi,yi ∈R
Commutators: [Xi,Yj] =δijZ.
Ordered basis: (Z,X1,Y1, . . . ,Xn,Yn).
Automorphism group
Let ω be the skew-symmetric bilinear form on hn specified by [A,B] =ω(A,B)Z, A,B∈hn.
Characterization (cf. [GoOnVi97])
A linear isomorphism ψ:hn→hn is a Lie algebra automorphism if and only if ψ·Z =cZ and ω(ψ·A, ψ·B) =cω(A,B) for some c 6= 0.
With respect to ordered basis: (Z,X1,Y1,X2,Y2, . . . ,Xn,Yn),
ω =
0 0 0 J
, where J=
0 1 0
−1 0 . ..
0 1
0 −1 0
.
Automorphism group
Proposition (cf. [Saal96, BiNa13])
The group of automorphisms Aut(hn) is given by r2 v
0 rg
, σ
r2 v 0 rg
: r >0,v∈R1×2n,g ∈Sp (n,R)
where
Sp (n,R) = n
g ∈R2n×2n : g>Jg =J o
is the n(2n+ 1)-dimensional symplectic group over R and
σ=
−1 0 · · · 0
0 0 1 0
1 0 ... . ..
0 1
0 0 1 0
.
Outline
1 Introduction
2 Classification
3 Isometry group
4 Geodesics
5 Conclusion
Williamson’s theorem & symplectic spectrum
Lemma (see, e.g., [dG06])
If M is a positive definite 2n×2n matrix, then there exists S ∈Sp (n,R) such that
S>M S =
λ1 0
λ1
λ2
λ2 . ..
λn
0 λn
where λ1 ≥λ2 ≥ · · · ≥λn>0.
Riemannian structures
Theorem ([BiNa13])
Any left-invariant Riemannian structure on Hn is isometric to exactly one of the structures
g1λ= 1 0
0 Λ
, Λ = diag(λ1, λ1, λ2, λ2, . . . , λn, λn) i.e., with orthonormal base
(Z,√1λ
1X1,√1λ
1Y1,√1λ
2X2,√1λ
2Y2, . . . ,√1
λnXn,√1
λnYn).
Here λ1≥λ2 ≥ · · · ≥λn>0 parametrize a family of (non-isometric) class representatives.
Riemannian structures
Proof sketch
g and g0 isometric if and only if g1(A,B) =g1(ψ·A, ψ·B) for some ψ∈Aut(h3).
In coordinates, application of inner automorphism:
ψ=
1 v 0 I2n
g01=ψ>g1ψ= 1
r2 0 0 Q0
.
Application of automorphism ψ0= diag(r2,r, . . . ,r):
g001=ψ0>g01ψ0 = 1 0
0 Q00
.
Application of symplectic transformations 1 0
0 g
, g ∈Sp(n,R) we can diagonalize to given form.
Sub-Riemannian structures
Theorem ([BiNa13])
Any left-invariant sub-Riemannian structure on Hn is isometric to exactly one of the structures (D,gλ) specified by
(D(1) = span(X1,Y1, . . . ,Xn,Yn)
gλ1 = Λ = diag(λ1, λ1, λ2, λ2, . . . , λn, λn) i.e., with orthonormal base
(√1λ
1X1,√1λ
1Y1,√1λ
2X2,√1λ
2Y2, . . . ,√1λ
nXn,√1λ
nYn).
Here 1 =λ1≥λ2 ≥ · · · ≥λn>0 parametrize a family of (non-isometric) class representatives.
Sub-Riemannian structures
Proof sketch
(D,g) and (D0,g0) isometric if and only if
ψ· D(1) =D0(1) and g1(A,B) =g1(ψ·A, ψ·B) for some ψ∈Aut(h3).
For any subspace s⊆hn, we have Lie(s)≤span(s,Z). So, if Lie(s) =hn and s6=hn, then s has codimension one and
s= span(X1+v1Z, . . . ,Xn+vnZ,Y1+vn+1Z, . . . ,Yn+v2nZ).
Accordingly, we have an inner automorphism
ψ=
1 −v 0 I2n
, ψ·s= span(X1, . . . ,Xn,Y1, . . . ,Yn).
Diagonalize metric by symplectic transformations.
Outline
1 Introduction
2 Classification
3 Isometry group
4 Geodesics
5 Conclusion
Structure of isometry group
The isometry group decomposes as semidirect product Iso(Hn,D,g) = HnoIso1(Hn,D,g)
of left translations (normal) and the isotropy group of the identity.
As Hn is simply connected, there is a one-to-one correspondence between Aut(Hn) and Aut(hn).
Accordingly, to determine Iso1(Hn,D,g), we need only find the subgroup of linearized isotropies
dIso1(Hn,D,g) =
ψ∈Aut(hn) : ψ· D(1) =D(1) g1(A,B) =g1(ψ·A, ψ·B)
.
Isotropy subgroup
Theorem
The group of linearized isotropies dIso1(Hn,D,gλ) for both the Riemannian and sub-Riemannian structures are given by
1 0 · · · 0
0 g1 0
... . ..
0 0 gk
, σ
1 0 · · · 0
0 g1 0
... . ..
0 0 gk
: gi ∈U (mi)
where the unitary group U (mi) = Sp (mi,R)∩O (2mi).
The numbers m1, . . . ,mk denote the multiplicities of the distinct values in the list (λ1, λ2, . . . , λn).
Isotropy subgroup
Proof sketch
Automorphisms:
ψ=
r2 v 0 rg
or ψ=σ
r2 v 0 rg
For the sub-Riemannian case, we have v = 0 as ψ· D(1) =D(1);
preservation of metric implies r = 1. Likewise for Riemannian case.
In either case: ψ= 1 0
0 g
, g>Jg =J, and g>Λg = Λ.
Can show that g must preserve eigenspaces of Λ2 and so g = diag(g1, . . . ,gk), g ∈GL (2mi,R).
Conditions g>Jg =J, and g>Λg = Λ then imply gi ∈Sp (mi,R) and gi ∈O(2mi), respectively.
Remarks
The automorphisms
ψ=
1 0 · · · 0
0 g1 0
... . ..
0 0 gn
, gi ∈SO (2)
are isotropies for all structures.
n≤dim Iso1(Hn,D,gλ) =Pk
i=1m2i ≤n2
Minimal symmetry: all values λ1, . . . , λn are distinct.
Maximal symmetry: all values λ1, . . . , λn are identical.
Riemannian structures not symmetric. However, sub-Riemannian structures are sub-symmetric.
Homotheties
Theorem
For the sub-Riemannian structures on Hn, a mapping φ: G→G is a diffeomorphism such that φ∗D=D and φ∗gλ =rgλ for some r >0 if and only if φ is the composition of an isometry and an automorphism δr ∈Aut(Hn) given by
δr :m(z,x,y)7→m(r z,√ r x,√
r y).
Remark
For the Riemannian structures, a mapping φ: G→G is an automorphism such that φ∗gλ=rgλ for some r >0 if and only if φ is an isometry (and r = 1).
Outline
1 Introduction
2 Classification
3 Isometry group
4 Geodesics
5 Conclusion
Sub-Riemannian geodesics
Theorem
The unit speed geodesics for the sub-Riemannian structure (Hn,D,gλ) are, up to composition with an isometry, given by
(i) g(t) =m(z(t),x(t),y(t)), where z(t) = 14
n
X
i=1 ci2 c02(2cλ0
i t−sin(2cλ0
i t))
xi(t) = cci
0sin(cλ0
it)
yi(t) = cci
0(1−cos(cλ0
it));
(ii) g(t) =m(0,x(t),0), where xi(t) = λci
it.
Here c1, . . . ,cn≥0, Pn i=1
ci2
λi = 1 and c0 >0 parametrize a family of geodesics.
Sub-Riemannian geodesics
Remark
A geodesic (from identity) is uniquely determined by its first and second derivative (at identity). For the above curves we have
˙
z(0) = 0 z¨(0) = 0
˙
xi(0) = λci
i x¨i(0) = 0
˙
yi(0) = 0 y¨i(0) = λci2 i
c0.
Sub-Riemannian geodesics
Proof sketch (1/3)
The length minimization problem
˙
g(t)∈ D(g(t)), g(0) =g0, g(t1) =g1, Z T
0
q
gλ( ˙g(t),g˙(t))→min
is equivalent to the energy minimization problem (or invariant optimal control problem)
˙
g(t) =g(t)
n
X
i=1
(uXi(t)Xi+uYi(t)Yi), g(0) =g0, g(T) =g1 Z T
0 n
X
i=1
λi(uXi(t)2+uYi(t)2)dt →min.
Sub-Riemannian geodesics
Proof sketch (2/3)
Via the Pontryagin Maximum Principle, lift problem to cotangent bundle T∗Hn= Hn×h∗n.
Geodesics are projections of integral curves of H(g,p) =
n
X
i=1 1 λi(p2X
i +pY2
i), (g,p)∈Hn×h∗n.
Accordingly, geodesics g(t) =m(z(t),x(t),y(t)) are solutions of
˙ z =
n
X
i=1 1
λixipYi x˙i = λ1
ipXi y˙i = λ1
ipYi
˙
pZ = 0 p˙Xi =−λ1
ipZpYi p˙Yi = λ1
ipZpXi.
Sub-Riemannian geodesics
Proof sketch (3/3)
By application of a left-translation, we may assume g(0) =1, i.e., z(0) =xi(0) =yi(0) = 0.
By application of an isotropy
ψ=
1 0 · · · 0
0 g1 0
... . ..
0 0 gn
, gi ∈SO (2)
we may assume ˙yi(0) = 0 and ˙xi(0)≥0, i.e., pYi(0) = 0 and pXi(0)≥0.
Finally, solve Cauchy problem and find unit speed reparametrization.
Sub-Riemannian geodesics
Totally geodesic submanifold N
Satisfies property: whenever a geodesic g(·) is tangent to N at some point g ∈N, then the entire trace of g(·) is contained in N.
Corollary
The subgroups with Lie algebra spanned by (Z,X1,Y1) (Z,X1,Y1,X2,Y2)
...
are totally geodesic submanifolds for (Hn,D,gλ).
Riemannian geodesics
Theorem
The unit speed geodesics for the Riemannian structure gλ are, up to a composition with an isometry, given by
(i) g(t) =m(z(t),x(t),y(t)), where z(t) =c0t+14
n
X
i=1 ci2 c02(2cλ0
i t−sin(2cλ0
i t))
xi(t) = cci
0sin(λc0
it)
yi(t) = cci
0(1−cos(cλ0
it));
(ii) g(t) =m(0,x(t),0), where xi(t) = λci
it if a0= 0.
Here c0,ci ≥0, c02+Pn i=1
ci2 λi = 1.
Riemannian geodesics
Remark
A geodesic (from identity) is uniquely determined by its first derivative (at identity). For the above curves we have
˙
z(0) =c0
˙
xi(0) = λci
i
˙
yi(0) = 0.
Relation between geodesics
Sub-Riemannian and Riemannian geodesics are very similar:
Sub-Riemannian geodesics Riemannian geodesics z(t) = 14Pci2
c02(2cλ0
i t−sin(2cλ0
i t))
xi(t) = cci
0sin(λc0
it)
yi(t) = cci
0(1−cos(cλ0
it))
z(t) =c0t+14Pci2 c02(2cλ0
i t−sin(2cλ0
i t))
xi(t) = cci
0sin(cλ0
it)
yi(t) = cci
0(1−cos(cλ0
it))
z(t) = 0 xi(t) = cci
0sin(λc0
it)
yi(t) = 0
z(t) = 0 xi(t) = cci
0sin(cλ0
it)
yi(t) = 0
Relation between geodesics
The mapping
π : Hn→R2n∼= Hn/Z(Hn), m(z,x,y)7→(x,y).
is a Lie group epimorphism with kernel kerπ = Z(Hn).
Let ˜G denote the collection of geodesics of (Hn,˜g).
Let G denote the collection of geodesics of (Hn,D,g).
Theorem
If (Hn,g)˜ tames (Hn,D,g) and D(1) = Z(hn)⊥ with respect to ˜g1,
then π( ˜G) =π(G).
Relation between geodesics
Let ˜g(·) be a geodesic of (Hn,˜g). There exists a unique curve g(·) on Hn such that
g(0) = ˜g(0), π(g(t)) =π(˜g(t)), g˙(t)∈ D(g(t)).
We call g(·) the D-projection of ˜g(·).
Corollary
If (Hn,˜g) tames (Hn,D,g) and D(1) = Z(hn)⊥ with respect to ˜g1, then the geodesics of (Hn,D,g) are exactly the D-projections of the geodesics of (Hn,g).˜
Outline
1 Introduction
2 Classification
3 Isometry group
4 Geodesics
5 Conclusion
Conclusion
Outlook
minimizing geodesics (cf. [TaYa04]) totally geodesic submanifolds
relation between geodesics: true for larger class?
affine distributions (& optimal control)
complete treatment for lower-dimensional Lie groups (cf. [AgBa12, Alm14, HaLe09, ArSa11, BoRo08, Maz12, MoSa10, MoAn02, Sac10])
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