Homogeneous groups of H-type

Homogeneous groups of
H-type
method of geometric mechanics
Irina Markina
University of Bergen
Norway
Homogeneous groups of H-type – p. 1/4

Contents:
• General definitions of homogeneous groups
of H-type
Homogeneous groups of H-type – p. 2/4

Contents:
• General definitions of homogeneous groups
of H-type
• Construction of groups satisfying the J 2
condition
Homogeneous groups of H-type – p. 2/4

Contents:
• General definitions of homogeneous groups
of H-type
• Construction of groups satisfying the J 2
condition
• Example: quaternion group
Homogeneous groups of H-type – p. 2/4

Definitions
Let
• G be a real Lie algebra,
Homogeneous groups of H-type – p. 3/4

Definitions
Let
• G be a real Lie algebra,
• 〈·, ·〉 scalar product,
Homogeneous groups of H-type – p. 3/4

Definitions
Let
• G be a real Lie algebra,
• 〈·, ·〉 scalar product,
• [·, ·] Lie brackets,
Homogeneous groups of H-type – p. 3/4

Definitions
Let
• G be a real Lie algebra,
• 〈·, ·〉 scalar product,
• [·, ·] Lie brackets,
• G = V1 ⊕ V2, orthogonal direct sum,
Homogeneous groups of H-type – p. 3/4

Definitions
Let
• G be a real Lie algebra,
• 〈·, ·〉 scalar product,
• [·, ·] Lie brackets,
• G = V1 ⊕ V2, orthogonal direct sum,
• [V1, V1] ⊆ V2 and [V1, V2] = [V2, V2] = 0.
Homogeneous groups of H-type – p. 3/4

Definitions
If there is a linear map J : V2 → End(V1)
• 〈JZX, X ′ 〉 = 〈Z, [X, X ′ ]〉,
∀ X, X ′ ∈ V1, ∀ Z ∈ V2,
Homogeneous groups of H-type – p. 4/4

Definitions
If there is a linear map J : V2 → End(V1)
• 〈JZX, X ′ 〉 = 〈Z, [X, X ′ ]〉,
∀ X, X ′ ∈ V1, ∀ Z ∈ V2,
• J T Z = −JZ, ∀ Z ∈ V2,
Homogeneous groups of H-type – p. 4/4

Definitions
If there is a linear map J : V2 → End(V1)
• 〈JZX, X ′ 〉 = 〈Z, [X, X ′ ]〉,
∀ X, X ′ ∈ V1, ∀ Z ∈ V2,
• J T Z = −JZ, ∀ Z ∈ V2,
• J 2 Z = −|Z|2 U, ∀ Z ∈ V2, U is the identity,
Homogeneous groups of H-type – p. 4/4

Definitions
If there is a linear map J : V2 → End(V1)
• 〈JZX, X ′ 〉 = 〈Z, [X, X ′ ]〉,
∀ X, X ′ ∈ V1, ∀ Z ∈ V2,
• J T Z = −JZ, ∀ Z ∈ V2,
• J 2 Z = −|Z|2 U, ∀ Z ∈ V2, U is the identity,
• we say G is H-type.
Homogeneous groups of H-type – p. 4/4

J 2 condition
G satisfies the J 2 condition if, whenever
X ∈ V1 and Z, Z ′ ∈ V2 with 〈Z, Z ′ 〉 = 0,
there exists Z ′′ in V2, such that
Homogeneous groups of H-type – p. 5/4

J 2 condition
G satisfies the J 2 condition if, whenever
X ∈ V1 and Z, Z ′ ∈ V2 with 〈Z, Z ′ 〉 = 0,
there exists Z ′′ in V2, such that
JZJZ ′X = JZ ′′X.
Homogeneous groups of H-type – p. 5/4

Classification
• R n 0, k ∈ N, is Euclidean space
Homogeneous groups of H-type – p. 6/4

Classification
• R n 0, k ∈ N, is Euclidean space
• G n 1 , n = 2k, k ∈ N, is Heisenberg algebra
Homogeneous groups of H-type – p. 6/4

Classification
• R n 0, k ∈ N, is Euclidean space
• G n 1 , n = 2k, k ∈ N, is Heisenberg algebra
• G n 3 , n = 4k, k ∈ N, is quaternion algebra
Homogeneous groups of H-type – p. 6/4

Classification
• R n 0, k ∈ N, is Euclidean space
• G n 1 , n = 2k, k ∈ N, is Heisenberg algebra
• G n 3 , n = 4k, k ∈ N, is quaternion algebra
• G 8 7 is octonion algebra.
Homogeneous groups of H-type – p. 6/4

Classification
• R n 0, k ∈ N, is Euclidean space
• G n 1 , n = 2k, k ∈ N, is Heisenberg algebra
• G n 3 , n = 4k, k ∈ N, is quaternion algebra
• G 8 7 is octonion algebra.
• Cowling M.; Dooley A. H.; Korányi A.; Ricci F. H-type groups and Iwasawa
decompositions. Adv. Math. 87 (1991), no. 1, 1–41.
Homogeneous groups of H-type – p. 6/4

Cayley-Dickson construction
• What are real numbers R?
Homogeneous groups of H-type – p. 7/4

Cayley-Dickson construction
• What are real numbers R?
• What are complex numbers C?
Homogeneous groups of H-type – p. 7/4

Cayley-Dickson construction
• What are real numbers R?
• What are complex numbers C?
• It is a pair (a, b) of real numbers a, b ∈ R,
Homogeneous groups of H-type – p. 7/4

Cayley-Dickson construction
• What are real numbers R?
• What are complex numbers C?
• It is a pair (a, b) of real numbers a, b ∈ R,
• a ∗ = a conjugate to a real number,
Homogeneous groups of H-type – p. 7/4

Cayley-Dickson construction
• What are real numbers R?
• What are complex numbers C?
• It is a pair (a, b) of real numbers a, b ∈ R,
• a ∗ = a conjugate to a real number,
• (a, b) ∗ = (a ∗ , −b) is conjugate to a complex
number,
Homogeneous groups of H-type – p. 7/4

Cayley-Dickson construction
• What are real numbers R?
• What are complex numbers C?
• It is a pair (a, b) of real numbers a, b ∈ R,
• a ∗ = a conjugate to a real number,
• (a, b) ∗ = (a ∗ , −b) is conjugate to a complex
number,
• multiplication (a, b)(c, d) = (ac − db ∗ , a ∗ d + cb).
Homogeneous groups of H-type – p. 7/4

Quaternions H
• It is a pair (a, b) of complex numbers a, b ∈ C,
Homogeneous groups of H-type – p. 8/4

Quaternions H
• It is a pair (a, b) of complex numbers a, b ∈ C,
• (a, b) ∗ = (a ∗ , −b) is conjugate to a quaternion,
Homogeneous groups of H-type – p. 8/4

Quaternions H
• It is a pair (a, b) of complex numbers a, b ∈ C,
• (a, b) ∗ = (a ∗ , −b) is conjugate to a quaternion,
• with the multiplication
(a, b)(c, d) = (ac − db ∗ , a ∗ d + cb).
Homogeneous groups of H-type – p. 8/4

Octonions O
• It is a pair (a, b) of quaternion numbers
a, b ∈ H,
Homogeneous groups of H-type – p. 9/4

Octonions O
• It is a pair (a, b) of quaternion numbers
a, b ∈ H,
• (a, b) ∗ = (a ∗ , −b) is conjugate to an octonion,
Homogeneous groups of H-type – p. 9/4

Octonions O
• It is a pair (a, b) of quaternion numbers
a, b ∈ H,
• (a, b) ∗ = (a ∗ , −b) is conjugate to an octonion,
• with the multiplication
(a, b)(c, d) = (ac − db ∗ , a ∗ d + cb).
Homogeneous groups of H-type – p. 9/4

Octonions O
• It is a pair (a, b) of quaternion numbers
a, b ∈ H,
• (a, b) ∗ = (a ∗ , −b) is conjugate to an octonion,
• with the multiplication
• . . .
(a, b)(c, d) = (ac − db ∗ , a ∗ d + cb).
Homogeneous groups of H-type – p. 9/4

Properties we loose
• R division algebra, associative, commutative,
self conjugate,
Homogeneous groups of H-type – p. 10/4

Properties we loose
• R division algebra, associative, commutative,
self conjugate,
• C division algebra, associative, commutative,
Homogeneous groups of H-type – p. 10/4

Properties we loose
• R division algebra, associative, commutative,
self conjugate,
• C division algebra, associative, commutative,
• H division algebra, associative,
Homogeneous groups of H-type – p. 10/4

Properties we loose
• R division algebra, associative, commutative,
self conjugate,
• C division algebra, associative, commutative,
• H division algebra, associative,
• O division algebra,
Homogeneous groups of H-type – p. 10/4

Properties we loose
• R division algebra, associative, commutative,
self conjugate,
• C division algebra, associative, commutative,
• H division algebra, associative,
• O division algebra,
• if xy = 0, then x = 0 or y = 0.
Homogeneous groups of H-type – p. 10/4

Unities
• R has only one 1 = (1, 0), 1 2 = 1.
Imaginary part has dimension 0.
Homogeneous groups of H-type – p. 11/4

Unities
• R has only one 1 = (1, 0), 1 2 = 1.
Imaginary part has dimension 0.
• C: 1 = (1, 0), 1 2 = 1 and i = (0, 1), i 2 = −1.
Imaginary part has dimension 1.
z = a + ib = Re z + i Im z, a, b ∈ R.
Homogeneous groups of H-type – p. 11/4

Unities
• H: 1 = (1, 0), 1 2 = 1 and
i1 = (i, 0), i2 = (0, 1), i3 = (0, i),
i 2 1 = i 2 2 = i 2 3 = −1.
Homogeneous groups of H-type – p. 12/4

Unities
• H: 1 = (1, 0), 1 2 = 1 and
i1 = (i, 0), i2 = (0, 1), i3 = (0, i),
i 2 1 = i 2 2 = i 2 3 = −1.
• q = a + i1b + i2c + i3d =
Re q + i1 Im1 q + i2 Im2 q + i3 Im3 q,
a, b, c, d ∈ R.
Homogeneous groups of H-type – p. 12/4

Unities
O: 1 = (1, 0), 1 2 = 1 and
with
j1 = (i1, 0), j2 = (i2, 0), j3 = (i3, 0),
j4 = (0, 1), j5 = (0, i1) j6 = (0, i2), j7 = (0, i3),
j 2 1 = j 2 2 = j 2 3 = j 2 4 = j 2 5 = j 2 6 = j 2 7 = −1.
Imaginary part has dimension 7.
Homogeneous groups of H-type – p. 13/4

R n (R n 0)
R n 0 = V1 ⊕ V2 with V1 = R n and V2 = ∅.
Homogeneous groups of H-type – p. 14/4

Heisenberg group G n 1 (C n ×R)
• z = (x, y) ∈ C, t ∈ R,
Homogeneous groups of H-type – p. 15/4

Heisenberg group G n 1 (C n ×R)
• z = (x, y) ∈ C, t ∈ R,
• w = (a, b) ∈ C, s ∈ R
Homogeneous groups of H-type – p. 15/4

Heisenberg group G n 1 (C n ×R)
• z = (x, y) ∈ C, t ∈ R,
• w = (a, b) ∈ C, s ∈ R
• (z, t) ◦ (w, s) = (z + w, t + s + Im zw ∗ ) =
(z + w, t + s + iz · w) = (z + w, t + s + J z · w),
Homogeneous groups of H-type – p. 15/4

Heisenberg group G n 1 (C n ×R)
• z = (x, y) ∈ C, t ∈ R,
• w = (a, b) ∈ C, s ∈ R
• (z, t) ◦ (w, s) = (z + w, t + s + Im zw ∗ ) =
(z + w, t + s + iz · w) = (z + w, t + s + J z · w),
where
J
0 −1
= .
1 0
Homogeneous groups of H-type – p. 15/4

Heisenberg group G n 1 (C n ×R)
• z = (x, y) ∈ C, t ∈ R,
• w = (a, b) ∈ C, s ∈ R
• (z, t) ◦ (w, s) = (z + w, t + s + Im zw ∗ ) =
(z + w, t + s + iz · w) = (z + w, t + s + J z · w),
where
J
0 −1
= .
1 0
Homogeneous groups of H-type – p. 15/4

Quaternion H-type group G n 3
(H n × R 3 )
• q ∈ H, t1, t2, t3 ∈ R,
Homogeneous groups of H-type – p. 16/4

Quaternion H-type group G n 3
(H n × R 3 )
• q ∈ H, t1, t2, t3 ∈ R,
• h ∈ H, s1, s2, s3 ∈ R
Homogeneous groups of H-type – p. 16/4

Quaternion H-type group G n 3
(H n × R 3 )
• q ∈ H, t1, t2, t3 ∈ R,
• h ∈ H, s1, s2, s3 ∈ R
• (q, t1, t2, t3) ◦ (h, s1, s2, s3) = (q + h,
Homogeneous groups of H-type – p. 16/4

Quaternion H-type group G n 3
(H n × R 3 )
• q ∈ H, t1, t2, t3 ∈ R,
• h ∈ H, s1, s2, s3 ∈ R
• (q, t1, t2, t3) ◦ (h, s1, s2, s3) = (q + h,
• t1 + s1 + Im1 qh ∗ , t2 + s2 + Im2 qh ∗ , t3 + s3 +
Im3 qh ∗ ) =
Homogeneous groups of H-type – p. 16/4

Quaternion H-type group G n 3
(H n × R 3 )
• q ∈ H, t1, t2, t3 ∈ R,
• h ∈ H, s1, s2, s3 ∈ R
• (q, t1, t2, t3) ◦ (h, s1, s2, s3) = (q + h,
• t1 + s1 + Im1 qh ∗ , t2 + s2 + Im2 qh ∗ , t3 + s3 +
Im3 qh ∗ ) =
• (q + h, t1 + s1 + i1q · h, t2 + s2 + i2q · h, t3 + s3 +
i3q · h) =
Homogeneous groups of H-type – p. 16/4

Quaternion H-type group G n 3
(q + h, t1 + s1 + J1q · h, t2 + s2 + J2q · h, t3 + s3 + J3q · h),
Homogeneous groups of H-type – p. 17/4

Quaternion H-type group G n 3
(q + h, t1 + s1 + J1q · h, t2 + s2 + J2q · h, t3 + s3 + J3q · h),
J1 =
⎡
⎢
⎣
0 1 0 0
−1 0 0 0
0 0 0 1
0 0 −1 0
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ , J2 = ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
0 0 0 −1
0 0 −1 0
0 1 0 0
1 0 0 0
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ , J3 = ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
0 0 −1 0
0 0 0 1
1 0 0 0
0 −1 0 0
⎤
⎥ .
⎥
⎦
Homogeneous groups of H-type – p. 17/4

Octonion H-type group G 1 7
(O × R 7 )
• p ∈ O, t = (t1, . . . , t7) ∈ R 7 ,
Homogeneous groups of H-type – p. 18/4

Octonion H-type group G 1 7
(O × R 7 )
• p ∈ O, t = (t1, . . . , t7) ∈ R 7 ,
• r ∈ O, s = (s1, . . . , s7) ∈ R 7 .
Homogeneous groups of H-type – p. 18/4

Octonion H-type group G 1 7
(O × R 7 )
• p ∈ O, t = (t1, . . . , t7) ∈ R 7 ,
• r ∈ O, s = (s1, . . . , s7) ∈ R 7 .
• (p, t) ◦ (r, s) = (p + r, t + s + Im pr ∗ ),
Homogeneous groups of H-type – p. 18/4

Octonion H-type group G 1 7
(O × R 7 )
• p ∈ O, t = (t1, . . . , t7) ∈ R 7 ,
• r ∈ O, s = (s1, . . . , s7) ∈ R 7 .
• (p, t) ◦ (r, s) = (p + r, t + s + Im pr ∗ ),
• where Im pr ∗ = (Im1 pr ∗ , . . . , Im7 pr ∗ ).
Homogeneous groups of H-type – p. 18/4

Octonion H-type group G 1 7
(O × R 7 )
• p ∈ O, t = (t1, . . . , t7) ∈ R 7 ,
• r ∈ O, s = (s1, . . . , s7) ∈ R 7 .
• (p, t) ◦ (r, s) = (p + r, t + s + Im pr ∗ ),
• where Im pr ∗ = (Im1 pr ∗ , . . . , Im7 pr ∗ ).
• (p, t) ◦ (r, s) = (p + r, t + s + Jp · r),
Homogeneous groups of H-type – p. 18/4

Octonion H-type group G 1 7
(O × R 7 )
• p ∈ O, t = (t1, . . . , t7) ∈ R 7 ,
• r ∈ O, s = (s1, . . . , s7) ∈ R 7 .
• (p, t) ◦ (r, s) = (p + r, t + s + Im pr ∗ ),
• where Im pr ∗ = (Im1 pr ∗ , . . . , Im7 pr ∗ ).
• (p, t) ◦ (r, s) = (p + r, t + s + Jp · r),
• where J = (j1, . . . , j7).
Homogeneous groups of H-type – p. 18/4

H-type group
• (0, 0) is the neutral element,
Homogeneous groups of H-type – p. 19/4

H-type group
• (0, 0) is the neutral element,
• (x, t) ↣ (−x, −t) is the inverse element.
Homogeneous groups of H-type – p. 19/4

H-type group
• (0, 0) is the neutral element,
• (x, t) ↣ (−x, −t) is the inverse element.
• If p = (x, t) and r = (y, s), then
is the left translation.
Lp(r) = p ◦ r
Homogeneous groups of H-type – p. 19/4

H-type algebras
The Lie algebra is associated with the left
invariant vector fields
X(p) = X(x, t) = Al(p)∂xl + Bm(p)∂tm:
X(p) = (Lp)∗X(0).
Homogeneous groups of H-type – p. 20/4

H-type algebras
The Lie algebra is associated with the left
invariant vector fields
X(p) = X(x, t) = Al(p)∂xl + Bm(p)∂tm:
X(p) = (Lp)∗X(0).
X(p) = d
f(p ◦ γ(s))|s=0,
ds
γ(0) = 0, d
ds γ(s)|s=0 = X(0).
Homogeneous groups of H-type – p. 20/4

Heisenberg algebra G n 1
• Basis of G n 1
X1 = ∂x1 − x2∂t, X2 = ∂x2
T = ∂t
+ x1∂t,
Homogeneous groups of H-type – p. 21/4

Heisenberg algebra G n 1
• Basis of G n 1
X1 = ∂x1 − x2∂t, X2 = ∂x2
T = ∂t
• [X1, X2] = X1X2 − X2X1 = 2T
+ x1∂t,
Homogeneous groups of H-type – p. 21/4

Heisenberg algebra G n 1
• Basis of G n 1
X1 = ∂x1 − x2∂t, X2 = ∂x2
T = ∂t
• [X1, X2] = X1X2 − X2X1 = 2T
• G n 1 = V1 ⊕ V2,
+ x1∂t,
V1 = span{X1, X2}, V2 = span{T }.
Homogeneous groups of H-type – p. 21/4

Quaternions algebra G n 3
• Basis of G n 3
X1(x, t) =∂x1 + + x2∂t1
X2(x, t) =∂x2 + − x1∂t1
X3(x, t) =∂x3 + + x4∂t1
X4(x, z) =∂x4 + − x3∂t1
− x4∂t2 − x3∂t3 ,
− x3∂t2 + x4∂t3 ,
+ x2∂t2 + x1∂t3 ,
+ x1∂t2 − x2∂t3 .
Homogeneous groups of H-type – p. 22/4

Quaternions algebra G n 3
• Basis of G n 3
X1(x, t) =∂x1 + + x2∂t1
X2(x, t) =∂x2 + − x1∂t1
X3(x, t) =∂x3 + + x4∂t1
X4(x, z) =∂x4 + − x3∂t1
• T1 = ∂t1 , T2 = ∂t2 , T3 = ∂t3 .
− x4∂t2 − x3∂t3 ,
− x3∂t2 + x4∂t3 ,
+ x2∂t2 + x1∂t3 ,
+ x1∂t2 − x2∂t3 .
Homogeneous groups of H-type – p. 22/4

Quaternions algebra G n 3
•
[X1, X2] = −2T1, [X1, X3] = 2T3, [X1, X4] = 2T2,
[X2, X3] = 2T2, [X2, X4] = −2T3, [X3, X4] = −2T1
Homogeneous groups of H-type – p. 23/4

Quaternions algebra G n 3
•
[X1, X2] = −2T1, [X1, X3] = 2T3, [X1, X4] = 2T2,
[X2, X3] = 2T2, [X2, X4] = −2T3, [X3, X4] = −2T1
• G n 3 = V1 ⊕ V2,
V1 = span{X1, X2, X3, X4},
V2 = span{T1, T2, T3}.
Homogeneous groups of H-type – p. 23/4

Horizontal curve
• A curve c(s) = (x(s), t(s)) is called horizontal
if ˙c(s) = 4
l=1 α(s)Xl(c(s)).
Homogeneous groups of H-type – p. 24/4

Horizontal curve
• A curve c(s) = (x(s), t(s)) is called horizontal
if ˙c(s) = 4
l=1 α(s)Xl(c(s)).
• A curve c(s) is horizontal if and only if
˙t1 = + x2 ˙x1 − x1 ˙x2 + x4 ˙x3 − x3 ˙x4 = J1x · ˙x,
˙t2 = − x4 ˙x1 − x3 ˙x2 + x2 ˙x3 + x1 ˙x4 = J2x · ˙x,
˙t3 = − x3 ˙x1 + x4 ˙x2 + x1 ˙x3 − x2 ˙x4 = J3x · ˙x,
where ˙x = ( ˙x1, . . . , ˙x4).
Homogeneous groups of H-type – p. 24/4

Hamiltonian formalism
∆h =
4
l=1
X 2 l = ∆x + |x| 2 ∆t +
3
m=1
Jmx · ∇x ∂tm
Homogeneous groups of H-type – p. 25/4

Hamiltonian formalism
∆h =
4
l=1
X 2 l = ∆x + |x| 2 ∆t +
3
m=1
Hamilton’s function is H(ξ, θ, x, t) =
= |ξ| 2 + |x| 2 |θ| 2 + Mx · ξ,
ξl = ∂xl , θm = ∂tm , and M = 3
m=1 θmJm.
Jmx · ∇x ∂tm
Homogeneous groups of H-type – p. 25/4

Hamiltonian system
⎧
⎪⎨
⎪⎩
˙x = ∂H
∂ξ
= 2ξ + Mx
˙tm = ∂H
∂θm = 2|x|2 θm + Jmx · ξ, m = 1, 2, 3
˙ξ = − ∂H
∂x = −2|θ|2 x + Mξ
˙θ = − ∂H
∂t
= 0.
Homogeneous groups of H-type – p. 26/4

Geodesics
Let P (x0, t0), Q(x1, t1) ∈ Gn 3. A geodesic between
P y Q is the projection of the solution of the
Hamiltonian system onto (x, t)-space, satisfying
x(0), t(0) = (x0, t0), x(1), t(1) = (x, t).
Homogeneous groups of H-type – p. 27/4

Geodesics
Let P (x0, t0), Q(x1, t1) ∈ Gn 3. A geodesic between
P y Q is the projection of the solution of the
Hamiltonian system onto (x, t)-space, satisfying
x(0), t(0) = (x0, t0), x(1), t(1) = (x, t).
Lemma. Any geodesic is a horizontal curve, but not all
horizontal curves are geodesics.
c(s) = ( s2
2
, s, s2
2
, s, s3
6 , c1, c2)
Homogeneous groups of H-type – p. 27/4

General solutions
x(s) =
1 − cos(2s|θ|)
2|θ| 2
t(s) =
¨x = 2M ˙x.
sin(2s|θ|)
M ˙x(0) + U ˙x(0),
2|θ|
θ| ˙x(0)|2
4|θ| 2 (s − sin(2s|θ|)
).
2|θ|
Homogeneous groups of H-type – p. 28/4

Geodesics (0, 0) ⇒ (x, 0)
A smooth curve c(s) is horizontal with constant
coordinates t if and only if
c(s) = (a1s, . . . , a4s, t1, t2, t3),
where a1, . . . , a4 ∈ R y a 2 1 + . . . + a 2 4 = 0.
Homogeneous groups of H-type – p. 29/4

Geodesics (0, 0) ⇒ (x, t)
Given a point Q(x, t) there is a finite number of
geodesics connecting O(0, 0) with Q. Let |θ| be a
solution of the equation
4|t|
|θ|
= µ(|θ|) =
|x| 2 sin2 |θ|
− cot |θ|.
Homogeneous groups of H-type – p. 30/4

Geodesics (0, 0) ⇒ (x, t)
xm(s) =
4 sin(2|θ|m) sin2 (s|θ|m) − sin(2s|θ|m) sin2
|θ|m
|x1 | 22|θ|m − sin(2|θ|m) T .
+ cot |θ|m sin(s|θ|m) cos(s|θ|m) + sin 2 (s|θ|m) U
tm(s) = t 2s|θ|m − sin(2s|θ|m)
2|θ|m − sin(2|θ|m)
x 1 ,
m = 1, . . . , N.
Homogeneous groups of H-type – p. 31/4

Finding |θ|
40
35
30
25
20
15
10
5
2.5 5 7.5 10 12.5 15 17.5
Figure 1: Solution of the equation 4|t|
|x| 2 = µ(|θ|)
Homogeneous groups of H-type – p. 32/4

Graphics of geodesics
Homogeneous groups of H-type – p. 33/4

Length of geodesics
l 2 = ν(|θ|)(|x| 2 + 4|t|),
Homogeneous groups of H-type – p. 34/4

Length of geodesics
l 2 = ν(|θ|)(|x| 2 + 4|t|),
14
12
10
8
6
4
2
2.5 5 7.5 10 12.5 15 17.5
Figure 3: The graph of ν(|θ|) =
|θ| 2
sin |θ|(sin |θ|−cos |θ|)+|θ|
Homogeneous groups of H-type – p. 34/4

geodesics (0, 0) ⇒ (0, t)
There is an infinite number of geodesics joining
O(0, 0) with Q(0, t) satisfying the equations
xm(s) =
1 − cos(2πms)
2πm|z 1 |
tm(s) = t s − sin(2πms)
T =
⎡
⎢
⎣
T ˙x(0) + sin(2πms)
2πm
2πm
0 t1 −t3 −t2
−t1 0 −t2 t3
t3 t2 0 t1
t2 −t3 −t1 0
, m ∈ N.
⎤
⎥
⎦ ,
U ˙x(0),
Homogeneous groups of H-type – p. 35/4

Graphs of geodesics
Figure 4: l 2 m = 4πm|t|, m ∈ N
Homogeneous groups of H-type – p. 36/4

Carnot-Carathéodory metric
• dC−C(P, Q) = inf{l(c) : l(c) is the length of
horizontal curve c joining P and Q}.
Homogeneous groups of H-type – p. 37/4

Carnot-Carathéodory metric
• dC−C(P, Q) = inf{l(c) : l(c) is the length of
horizontal curve c joining P and Q}.
• Let M be two step group, then
Homogeneous groups of H-type – p. 37/4

Carnot-Carathéodory metric
• dC−C(P, Q) = inf{l(c) : l(c) is the length of
horizontal curve c joining P and Q}.
• Let M be two step group, then
• if (M, dC−C) is complete, then it is possible to
join any two points by geodesics,
Homogeneous groups of H-type – p. 37/4

Carnot-Carathéodory metric
• dC−C(P, Q) = inf{l(c) : l(c) is the length of
horizontal curve c joining P and Q}.
• Let M be two step group, then
• if (M, dC−C) is complete, then it is possible to
join any two points by geodesics,
• if there is a point P such that any geodesic
starting from P can be continued infinitely,
then (M, dC−C) is complete,
Homogeneous groups of H-type – p. 37/4

Carnot-Carathéodory metric
• any nonconstant geodesic locally coincides
with the shortest curve,
Homogeneous groups of H-type – p. 38/4

Carnot-Carathéodory metric
• any nonconstant geodesic locally coincides
with the shortest curve,
• any shortest curve is a geodesic.
Homogeneous groups of H-type – p. 38/4

Carnot-Carathéodory metric
• any nonconstant geodesic locally coincides
with the shortest curve,
• any shortest curve is a geodesic.
• Corollary. The metric space (G n 3, dC−C) is complete
and the shortest curve is geodesic.
Homogeneous groups of H-type – p. 38/4

C-C metric and geodesics
dC−C(O, Q) = {l(c) : c
is the shortest geodesic joining O and Q}.
d 2 C−C (O, Q) = ν(|θ|1)(|x| 2 + 4|t|), where |θ|1
is the smallest solution of the equation
4|t|
|x| 2 = µ(|θ|).
Homogeneous groups of H-type – p. 39/4

Some nice properties
Let c(s) be a geodesic. Then for any q ∈ G n 3
• ˜c(s) = q ◦ c(c) is also a geodesic,
Homogeneous groups of H-type – p. 40/4

Some nice properties
Let c(s) be a geodesic. Then for any q ∈ G n 3
• ˜c(s) = q ◦ c(c) is also a geodesic,
• the geodesics c(s) and ˜c(s) have the same
length,
Homogeneous groups of H-type – p. 40/4

Some nice properties
Let c(s) be a geodesic. Then for any q ∈ G n 3
• ˜c(s) = q ◦ c(c) is also a geodesic,
• the geodesics c(s) and ˜c(s) have the same
length,
• the vectors ˙c(s) and ¨c(s) are orthogonal,
Homogeneous groups of H-type – p. 40/4

Some nice properties
Let c(s) be a geodesic. Then for any q ∈ G n 3
• ˜c(s) = q ◦ c(c) is also a geodesic,
• the geodesics c(s) and ˜c(s) have the same
length,
• the vectors ˙c(s) and ¨c(s) are orthogonal,
• the lengths of ˙c(s) and ¨c(s) are constant along
the geodesic,
Homogeneous groups of H-type – p. 40/4

Some nice properties
Let c(s) be a geodesic. Then for any q ∈ G n 3
• ˜c(s) = q ◦ c(c) is also a geodesic,
• the geodesics c(s) and ˜c(s) have the same
length,
• the vectors ˙c(s) and ¨c(s) are orthogonal,
• the lengths of ˙c(s) and ¨c(s) are constant along
the geodesic,
• the curvature of geodesics is constant and
κ(s) =
d ˙c
ds | ˙c| = 2|θ|
Homogeneous groups of H-type – p. 40/4

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The end
Homogeneous groups of H-type – p. 42/4