Geodesic completeness of the left-invariant metrics on ${{\mathbb{R}} H^n} $
UDC 514 We give the full classification of left-invariant metrics of an arbitrary signature on the Lie group corresponding to the real hyperbolic space. We show that all metrics have constant sectional curvature and that they are geodesically complete only in the Riemannian case.
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| author | Vukmirović, Srdjan Šukilović, Tijana Vukmirović, Srdjan Šukilović, Tijana Vukmirović, Srdjan Šukilović, Tijana |
| author_facet | Vukmirović, Srdjan Šukilović, Tijana Vukmirović, Srdjan Šukilović, Tijana Vukmirović, Srdjan Šukilović, Tijana |
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| description | UDC 514
We give the full classification of left-invariant metrics of an arbitrary signature on the Lie group corresponding to the real hyperbolic space. We show that all metrics have constant sectional curvature and that they are geodesically complete only in the Riemannian case. |
| doi_str_mv | 10.37863/umzh.v72i5.645 |
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DOI: 10.37863/umzh.v72i5.645
UDC 514
S. Vukmirović, T. Šukilović (Univ. Belgrade, Serbia)
GEODESIC COMPLETENESS OF THE LEFT-INVARIANT METRICS ON \BbbR \bfitH \bfitn *
ГЕОДЕЗИЧНА ПОВНОТА ЛIВОIНВАРIАНТНИХ МЕТРИК НА \BbbR \bfitH \bfitn
We give the full classification of left-invariant metrics of an arbitrary signature on the Lie group corresponding to the real
hyperbolic space. We show that all metrics have constant sectional curvature and that they are geodesically complete only
in the Riemannian case.
Наведено повну класифiкацiю лiвоiнварiантних метрик довiльної сигнатури на групi Лi, що вiдповiдає дiйсному
гiперболiчному просторовi. Показано, що всi такi метрики мають сталу кривизну перерiзу i геодезично повнi лише
в рiмановому випадку.
It is a well-known fact that the real hyperbolic space \BbbR Hn with the standard Riemannian metric
of constant negative curvature has a structure of a Lie group such that the metric is left-invariant.
Milnor [8] considered a special class of solvable Lie groups with the property that the commutator
[x, y] is a linear combination of x and y for any two elements from the corresponding Lie algebra.
Moreover, he has shown that such an algebra is isomorphic to the Lie algebra of \BbbR Hn and that every
left-invariant Riemannian metric on such group has a constant negative sectional curvature. In the
Lorentz case, Wolf [13] showed that this group admits a flat metric, while Nomizu [9] proved that for
an arbitrary K \in \BbbR there exists a left-invariant metric with sectional curvature K. In [10], Nomizu
considered in detail these metrics that he called Lorentz – Poincaré metrics.
In this paper we classify all left-invariant metrics of arbitrary signature on the Lie group cor-
responding to the hyperbolic space. This problem has also been considered in [6]. According to
Arnold [1], geodesic of an arbitrary left invariant metric on a Lie group G can be seen as a motion
of a “generalized rigid body” with a configuration space G. In the Riemannian case all geodesic are
complete, but Gudeiri [4] gave an example of the Lorentzian metric on four-dimensional Lie group
with non-complete geodesics. Lauret [7] has classified all Riemannian left-invariant metrics on the
four-dimensional nilpotent Lie groups, while the authors have generalized that result to an arbitrary
signature [2]. Calvaruso [3] has classified Lorentzian left-invariant metrics on the four-dimensional
Lie groups that are Einstein or Ricci-parallel. In [11] the Riemannian and Lorentzian left-invariant
metrics on the Heisenberg – Lie group were classified.
In the preliminary section we introduce a basic notation and give an explanation what does it
mean to classify the left-invariant metrics.
In the Theorem 2.2 we present classification of the arbitrary signature left-invariant metrics on
the Lie group \BbbR Hn. We show that in the Riemannian case the only metrics are the standard metrics
of constant curvature K < 0 of hyperbolic space. In the Lorentz case, every metric from our
classification is isometric either to the flat metric obtained by Wolf, or to the metric of constant
curvature K \not = 0 obtained by Nomizu. Also, in the Theorem 2.3 we prove that every non-flat metric
on \BbbR Hn is a metric of the constant sectional curvature.
* This research was supported by the Serbian Ministry of Science (Project 174012, Geometry, Education and Visuali-
zation with Applications).
c\bigcirc S. VUKMIROVIĆ, T. ŠUKILOVIĆ, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5 611
612 S. VUKMIROVIĆ, T. ŠUKILOVIĆ
The geodesical completeness of metrics is considered in Section 3. We show that all geodesic
curves are complete only in the Riemannian case (Theorems 3.1 and 3.2).
In Section 4, we exhibit isometric imbedding of \BbbR Hn into the space forms of the curvature K.
1. Preliminaries. The group structure on a half-space model of a real hyperbolic space \BbbR Hn =
= \{ (x1, . . . , xn) \in \BbbR n| xn > 0\} is given by
(x1, . . . , xn - 1, xn) (y1, . . . , yn - 1, yn) =
\bigl(
x1 + xny1, . . . , xn - 1 + xnyn - 1, xnyn
\bigr)
. (1)
Denote by \{ e1, . . . , en\} a corresponding basis of the Lie algebra rn = \mathrm{L}\mathrm{i}\mathrm{e} \BbbR Hn. It is a semidirect
product \BbbR en \ltimes n, where n = \scrL (e1, . . . , en - 1) is an Abelian ideal and \mathrm{a}\mathrm{d}(en) | n = \mathrm{i}\mathrm{d}, i.e., the non-
null commutators are
[en, ek] = ek, k < n.
Denote by \scrS (rn) a set of non-equivalent inner products of an algebra rn. With a basis of the
Lie algebra rn fixed, the set \scrS (rn) is identified with symmetric matrices S of an arbitrary signature
modulo the following action of the automorphism group:
S \mapsto \rightarrow F TSF, F \in \mathrm{A}\mathrm{u}\mathrm{t} (rn). (2)
Here we denoted by \mathrm{A}\mathrm{u}\mathrm{t} (rn) the group of automorphisms of the Lie algebra rn that is defined by
\mathrm{A}\mathrm{u}\mathrm{t} (rn) :=
\bigl\{
F : rn \rightarrow rn | F linear, bijective, [Fx, Fy] = F [x, y], x, y \in rn
\bigr\}
.
It is easy to check the following lemma.
Lemma 1.1. The group \mathrm{A}\mathrm{u}\mathrm{t} (rn) of automorphisms of Lie algebra rn, in basis \{ e1, e2, . . . , en\} ,
consists of real matrices of form
\mathrm{A}\mathrm{u}\mathrm{t} (rn) =
\Biggl\{ \Biggl(
A a
0 1
\Biggr) \bigm| \bigm| \bigm| \bigm| \bigm| A \in GL(n - 1,\BbbR ), a \in \BbbR n - 1
\Biggr\}
\sim = \mathrm{A}ffn - 1(\BbbR ), (3)
i.e., it is isomorphic to a group of affine transformations of \BbbR n - 1.
Let x \in \BbbR Hn with x = (x1, . . . , xn). For the left translations Lx the differential dLx in every
point y \in \BbbR Hn is given by
(dLx)
\biggl(
\partial
\partial xk
(y)
\biggr)
= xn
\partial
\partial xk
Lx(y).
Therefore, the left-invariant vector fields XL
1 , . . . , X
L
n are given by
XL
k (x) = xn
\partial
\partial xk
, k \leq n. (4)
2. Left-invariant metrics. Let us denote by Ip,r the diagonal matrix \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\epsilon 1, \epsilon 2, . . . , \epsilon p+r),
where \epsilon k = - 1, 1 \leq k \leq p, and \epsilon k = 1, p+ 1 \leq k \leq p+ r.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
GEODESIC COMPLETENESS OF THE LEFT-INVARIANT METRICS ON \BbbR Hn 613
Theorem 2.1. The set \scrS (rn) of non-equivalent inner products of an arbitrary signature on the
algebra rn is represented by the following matrices:
S\lambda =
\Biggl(
Ip,r 0
0 \lambda
\Biggr)
, p+ r = n - 1, \lambda \not = 0,
S0 =
\left(
0 0 1
0 Ip,r 0
1 0 0
\right) , p+ r = n - 2.
Proof. Let F \in \mathrm{A}\mathrm{u}\mathrm{t} (rn) and denote by \=S the arbitrary symmetric matrix representing the inner
product q in the basis \{ e1, . . . , en\} . In the same basis F is represented by the matrix (3). We are
looking for the new basis such that F T \=SF has the simplest form.
If we represent \=S in the following form:
\=S =
\Biggl(
S v
vT s
\Biggr)
,
where S = ST is (n - 1)\times (n - 1) matrix, v \in \BbbR n - 1 and s \in \BbbR , then
F T \=SF =
\Biggl(
ATSA AT (Sa+ v)
(aTS + vT )A aTSa+ vTa+ aT v + s
\Biggr)
.
Now, we distinguish between two cases.
Case 1. S is a regular matrix of signature (p, r). Since S is symmetric there exists A \in GL(n -
- 1,\BbbR ) such that ATSA = Ip,r, p+ r = n - 1. Setting a = - S - 1v, we get that the corresponding
inner product is S\lambda , with \lambda = s - vTS - 1v. Since \=S is a non-singular matrix, \lambda \not = 0 must hold.
Case 2. If S is not regular, without loss of generality, we can assume that S has the form\Biggl(
0 0
0 \~S
\Biggr)
,
where \~S is a regular matrix of signature (p, r), p+ r = n - 2. Then there exists a regular matrix \widetilde A
such that
A =
\left( 1
w
0
0 \widetilde A
\right) and ATSA =
\Biggl(
0 0
0 Ip,r
\Biggr)
.
For the vector v = (w, \=vT )T , w \not = 0, \=v \in \BbbR n - 2, we set a = (a1, \=a
T )T :
a1 =
\=vT \~S - 1\=v - s
2w
\in \BbbR , \=a = - \~S - 1\=v \in \BbbR n - 2,
to obtain the inner product S0.
Theorem 2.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
614 S. VUKMIROVIĆ, T. ŠUKILOVIĆ
The inner product q on rn gives rise to a left-invariant metric g on the corresponding Lie group.
For global coordinates (x1, . . . , xn) on \BbbR Hn, using the left-invariant vector fields (4), we can find a
coordinate description of metrics defined in the previous theorem.
Theorem 2.2. Each left-invariant metric on the group \BbbR Hn, up to an automorphism of \BbbR Hn,
is isometric to one of the following:
g\lambda =
1
x2n
\bigl(
- dx21 - . . . - dx2p + dx2p+1 + . . .+ dx2n - 1 + \lambda dx2n
\bigr)
, \lambda \not = 0,
g0 =
1
x2n
\bigl(
- dx22 - . . . - dx2p+1 + dx2p+2 + . . .+ dx2n - 1 + 2dx1dxn
\bigr)
.
Wolf [13] showed that \BbbR Hn admits flat metrics and, in the four-dimensional case, according to
classification of Jensen [5], we know that it also admits Einstein metrics. Later, Milnor [8] has shown
that every left-invariant positive definite metric on \BbbR Hn has a negative sectional curvature, while
Nomizu [9] proved that for every K \in \BbbR one can find the left-invariant Lorentz metric on \BbbR Hn
with K as constant sectional curvature. Yet, we are able to prove more.
Theorem 2.3. All left-invariant metrics of an arbitrary signature on \BbbR Hn have a constant
sectional curvature.
Proof. In order to prove the theorem we need to calculate the curvature tensor. We use the
identification of the left-invariant vector fields XL
k with their value in the unit element XL
k (e) = ek.
Recall that the curvature operators R(ei, ej) belong to the algebra \mathrm{s}\mathrm{o}(q) preserving the inner
product q, i.e.,
\mathrm{s}\mathrm{o}(q) :=
\bigl\{
A \in gl(rn)
\bigm| \bigm| AS + SAT = 0
\bigr\}
,
where S denotes the matrix of q. This algebra can be identified with the space \Lambda 2rn of bivectors,
whose action on rn is given by
(x \wedge y)z := q(y, z)x - q(x, z)y, x, y, z \in rn.
By using standard calculations, for the metric g\lambda , we get that the connection is given by the
non-zero expressions
\nabla eiei =
\epsilon i
\lambda
en, \epsilon i \in \{ - 1, 1\} , \nabla eien = - ei, i < n, (5)
and the curvature operators are given by
R(ei, ej) = - 1
\lambda
ei \wedge ej .
From the previous is apparent that the sectional curvature is constant K = - 1
\lambda
.
For metric g0 all components of curvature tensor R vanish, thus the metric is flat and K = 0.
Theorem 2.3 is proved.
3. Geodesics. Every C1 curve c(t) on the Lie group G, up to the left translations, gives rise to
the curve
\gamma (t) = L - 1
c(t)\ast \.c(t) (6)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
GEODESIC COMPLETENESS OF THE LEFT-INVARIANT METRICS ON \BbbR Hn 615
on the corresponding Lie algebra g. The curves of g associated to geodesics are solutions of the
equations
\.x = \mathrm{a}\mathrm{d}\ast x x, (7)
where \mathrm{a}\mathrm{d}\ast x stands for the adjoint of \mathrm{a}\mathrm{d}x relative to the inner product on g.
First, let us consider the Einstein metric g\lambda on \BbbR Hn. Fixing the basis \{ e1, . . . , en\} \in rn, an easy
computation gives us
\mathrm{a}\mathrm{d}\ast ekek = - \epsilon k
\lambda
en, \mathrm{a}\mathrm{d}\ast enek = ek, k < n.
In local coordinates (x1, . . . , xn), for \gamma (t) =
\sum n
k=1
xk(t)ek, from the equation (7) we obtain the
system
\.xk = xkxn, k < n, \.xn = - 1
\lambda
n - 1\sum
j=1
\epsilon jx
2
j . (8)
Let C1, . . . , Cn \in \BbbC and let us denote by C2
n+1 = - 1
\lambda
\sum n - 1
k=1
\epsilon kC
2
k . Then the solutions \gamma (t) =
= (x1(t), . . . , xn(t)) of the system (8) are given by
xk(t) =
Ck
\mathrm{c}\mathrm{o}\mathrm{s}(Cn+1t+ Cn)
, k < n,
xn(t) = Cn+1 \mathrm{t}\mathrm{a}\mathrm{n}(Cn+1t+ Cn), for C2
n+1 \not = 0, 1,
(9)
xk(t) =
Ck
t+ Cn
, k < n, xn(t) = - 1
t+ Cn
, for C2
n+1 = 1, (10)
xk(t) = Cke
Cnt, k < n, xn(t) = Cn, for Cn+1 = 0. (11)
Note that constants C1, . . . , Cn must be real in case of the solutions (10) and (11). For the solution (9)
they can be either real or complex, but they need to satisfy the additional constraints which will be
explained in detail in the proof of the following theorem.
Theorem 3.1. The left-invariant metric g\lambda on \BbbR Hn is geodesically complete if and only if it is
positive definite.
Proof. First, note that because of the left-invariance we may consider only the curves \gamma (t) in
the Lie algebra rn defined by (6).
A geodesic curve whose tangent vector is en corresponds to the solution (11) with Ck = 0,
k < n, and Cn = 1. Those are the vertical lines ending on the hyperplane xn = 0 and they are
complete in every signature.
Let v = \gamma (0) \not = en be a tangent vector of a geodesic curve. Notice that from (5) follows that
the two-dimensional plane \alpha = \scrL (v, en) is totally geodesic. Therefore, it is enough to discuss the
induced signature in that plane.
It is not difficult to calculate that for the solution (9) | v| 2 = - \lambda C2
n+1, for (10) | v| 2 = 0 and in
the last case (11) | v| 2 = \lambda C2
n. If the plane \alpha is non-degenerate then we consider solutions (9) and
(10), while the solution (11) occurs only if \alpha is degenerate.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
616 S. VUKMIROVIĆ, T. ŠUKILOVIĆ
Case 1. Suppose that the plane \alpha is Riemannian. Then C2
n+1 < 0, i.e., Cn+1 = iD, D \in \BbbR . In
order to determine the constant Cn we must consider the Gram determinant associated to the plane
\alpha . One can calculate that
G = - \lambda 2C2
n+1(1 + \mathrm{t}\mathrm{a}\mathrm{n}2Cn). (12)
In Riemannian case G must be positive, which yields Cn = iC, C \in \BbbR . Note that in order to obtain
the real solutions, all the other constants Ck, k < n, must be real. The trigonometric functions in
(9) become hyperbolic functions \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} and - \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}, so the curves are complete. It is easy to check
that the corresponding geodesic curves on the Lie group \BbbR Hn are half-ellipses with centers on the
hyperplane xn = 0. The plane \alpha is isometric to the standard hyperbolic plane.
Case 2. Suppose that the plane \alpha is Lorentzian.
If | v| 2 \not = 0, then the solution is given by (9) and we distinguish between two cases.
When the vectors v and en are of the same character, i.e., of the same signature, then C2
n+1 < 0.
Therefore, we must take Cn+1 = iD, D \in \BbbR . Here, the Gram determinant (12) must be negative,
thus Cn = iC +
\pi
2
, C \in \BbbR . Also, all the constants Ck, k < n, must be purely imaginary. The
trigonometric functions in (9) become hyperbolic functions \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} and - \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h} .
When the vectors v and en are of the different character, then C2
n+1 > 0 and we have the
solution (9) where all the constants are real. In both cases the geodesics are incomplete. The
corresponding geodesic curves on \BbbR Hn are branches of hyperbolas satisfying xn > 0.
If | v| 2 = 0, then the corresponding curves are given by (10). In \BbbR Hn those are the straight lines
ending on the hyperplane xn = 0 and they are geodesically incomplete.
Case 3. Suppose that the plane \alpha is degenerate. The vector u = v - Cnen is a null vector
orthogonal to all vectors from \alpha . The corresponding solutions are complete geodesics given by (11).
These are parabolas on \BbbR Hn.
We can conclude that the metric is complete if and only if for every tangent vector v the corres-
ponding plane \alpha is Riemannian.
Theorem 3.1 is proved.
Similarly, for the flat metric g0, in local coordinates, we have the system
\.x1 = - x1xn -
n - 1\sum
j=2
\epsilon jx
2
j , \.xk = xkxn, 1 < k \leq n.
The solutions to the system above are given by
x1(t) = C1(t+ Cn) +
C0
2(t+ Cn)
,
xk(t) =
Ck
t+ Cn
, 2 \leq k < n, xn(t) = - 1
t+ Cn
,
x1(t) = C1 - tC0, xk(t) = Ck, 2 \leq k < n, xn(t) = 0,
with C0 =
\sum n - 1
k=2
\epsilon kC
2
k and Ck \in \BbbR , k \leq n.
Consequently, the following theorem holds.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
GEODESIC COMPLETENESS OF THE LEFT-INVARIANT METRICS ON \BbbR Hn 617
Theorem 3.2. The pseudo-Riemannian metric g0 on the Lie group \BbbR Hn is geodesically incom-
plete.
4. Isometric imbedding into the space forms. Denote by \BbbR n
p the space \BbbR n with the pseudo-
Riemannian metric g(X,Y ) = -
\sum p
k=1
xkyk +
\sum n
k=p+1
xkyk for every X,Y \in \BbbR n.
Let Sn
p \subseteq \BbbR n+1
p be the de Sitter space
Sn
p =
\bigl\{
u = (u0, . . . , un)
\bigm| \bigm| - u20 - . . . - u2p - 1 + u2p + . . .+ u2n = - \lambda , \lambda < 0
\bigr\}
.
This is the hypersurface in \BbbR n+1
p with its induced metric of signature (p, n - p) of constant sectional
curvature K = - \lambda - 1 > 0.
Similarly, denote by Hn
p \subseteq \BbbR n+1
p+1 the anti-de Sitter space
Hn
p =
\bigl\{
u = (u0, . . . , un)
\bigm| \bigm| - u20 - . . . - u2p + u2p+1 + . . .+ u2n = - \lambda , \lambda > 0
\bigr\}
with its induced metric of signature (p, n - p) and constant sectional curvature K = - \lambda - 1 < 0.
Define \~Sn
p and \~Hn
p to be the respective connected, simply connected manifolds corresponding to
Sn
p and Hn
p .
According to Wolf [12] every complete connected pseudo-Riemannian manifold of signature
(p, n - p) and constant sectional curvature K has an universal pseudo-Riemannian covering \~Sn
p if
K > 0, \~Hn
p if K < 0, and \BbbR n
p if K = 0. Our metrics g\lambda and g0 have constant sectional curvature
and, although they are not always complete, we are interested in finding a local isometry into the
space forms.
Theorem 4.1. (\BbbR Hn, g\lambda ) of signature (p, n - p) is isometric to the part of Sn
p (if \lambda < 0) and
Hn
p (if \lambda > 0) determined by the condition u0 + un > 0.
Proof. Suppose that \lambda < 0. Then the metric g\lambda has the form
g\lambda =
1
x2n
\bigl(
- dx21 - . . . - dx2p - 1 + dx2p + . . .+ dx2n - 1 + \lambda dx2n
\bigr)
=
=
1
x2n
\Biggl(
n - 1\sum
k=1
\epsilon kx
2
k + \lambda x2n
\Biggr)
, \lambda < 0.
We define an isometric imbedding f : \BbbR Hn \rightarrow Sn
p by
f(x) = f(x1, . . . , xn) = (u0, u1, . . . , un) = u,
where
u0 =
1 +
\Bigl( \sum n - 1
k=1
\epsilon kx
2
k + \lambda x2n
\Bigr)
2xn
, uk =
xk
xn
, 1 \leq k < n,
un =
1 -
\Bigl( \sum n - 1
k=1
\epsilon kx
2
k + \lambda x2n
\Bigr)
2xn
.
The image f(\BbbR Hn) is an open submanifold\bigl\{
u = (u0, . . . , un) \in Sn
p
\bigm| \bigm| u0 + un > 0
\bigr\}
The proof of the case when \lambda > 0 is similar, only replacing Sn
p with Hn
p .
Theorem 4.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
618 S. VUKMIROVIĆ, T. ŠUKILOVIĆ
Remark 4.1. The previous theorem has been proven by Nomizu [9] in the Lorentz case. Fol-
lowing the reasoning from the same paper, one can show that there exists an isomorphism h :
\BbbR Hn \rightarrow SO+(p, n - p) such that the mapping f is equivariant, meaning that the following dia-
gram commutes for every g \in \BbbR Hn :
\BbbR Hn Lg - - - - \rightarrow \BbbR Hn \downarrow f
\downarrow f
Sn
p
h(g) - - - - \rightarrow Sn
p\bigl(
the same holds if we replace Sn
p with Hn
p
\bigr)
.
Remark 4.2. Note that a geodesic curve c(t) in \BbbR Hn is incomplete if and only if f(c(t)) reaches
the boundary u0 + un = 0 for a finite value of the affine parameter t. In the Figure 1 we illustrate
this with the example of geodesics on \BbbR H2.
Fig. 1. Geodesics on \BbbR H2 : Riemannian case (left), Lorentz case (right).
Theorem 4.2.
\bigl(
\BbbR Hn, g0
\bigr)
of signature (p, n - p) is isometric to the part of \BbbR n
p determined by
the condition y1 + yn > 0.
Proof. We can define the following change of coordinates, i.e., the map from \BbbR Hn \subset \BbbR n
p
to \BbbR n
p :
y1 =
1 +
\Bigl(
2x1xn +
\sum n - 1
k=2
\epsilon kx
2
k
\Bigr)
2xn
, yk =
xk
xn
, 1 < k < n,
yn =
1 -
\Bigl(
2x1xn +
\sum n - 1
k=2
\epsilon kx
2
k
\Bigr)
2xn
.
In this new coordinates, the metric g0 has the form
g\prime 0 = - dy21 +
n - 1\sum
k=2
\epsilon kdy
2
k + dy2n.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
GEODESIC COMPLETENESS OF THE LEFT-INVARIANT METRICS ON \BbbR Hn 619
This is a part of an open half-space of the flat space form satisfying the relation y1 + yn > 0.
Theorem 4.2 is proved.
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Received 07.04.17,
after revision — 18.12.19
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
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| id | umjimathkievua-article-645 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English Ukrainian |
| last_indexed | 2026-03-24T02:03:28Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/73/efe038cfc598bf48914f846cc6f02773.pdf |
| spelling | umjimathkievua-article-6452022-03-26T11:01:39Z Geodesic completeness of the left-invariant metrics on ${{\mathbb{R}} H^n} $ Geodesic completeness of the left-invariant metrics on ${{\mathbb{R}} H^n} $ Vukmirović, Srdjan Šukilović, Tijana Vukmirović, Srdjan Šukilović, Tijana Vukmirović, Srdjan Šukilović, Tijana UDC 514 We give the full classification of left-invariant metrics of an arbitrary signature on the Lie group corresponding to the real hyperbolic space.&nbsp;We show that all metrics have constant sectional curvature and that they are geodesically complete only in the Riemannian case. Наведено повну класифікацію лівоінваріантних метрик довільної сигнатури на групі Лі, що відповідає дійсному гіперболічному просторові.&nbsp;Показано, що всі такі метрики мають сталу кривизну перерізу і геодезично повні лише в рімановому випадку. Institute of Mathematics, NAS of Ukraine 2020-03-29 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/645 10.37863/umzh.v72i5.645 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 5 (2020); 611–619 Український математичний журнал; Том 72 № 5 (2020); 611–619 1027-3190 en uk https://umj.imath.kiev.ua/index.php/umj/article/view/645/8687 https://umj.imath.kiev.ua/index.php/umj/article/view/645/8688 |
| spellingShingle | Vukmirović, Srdjan Šukilović, Tijana Vukmirović, Srdjan Šukilović, Tijana Vukmirović, Srdjan Šukilović, Tijana Geodesic completeness of the left-invariant metrics on ${{\mathbb{R}} H^n} $ |
| title | Geodesic completeness of the left-invariant metrics on ${{\mathbb{R}} H^n} $ |
| title_alt | Geodesic completeness of the left-invariant metrics on ${{\mathbb{R}} H^n} $ |
| title_full | Geodesic completeness of the left-invariant metrics on ${{\mathbb{R}} H^n} $ |
| title_fullStr | Geodesic completeness of the left-invariant metrics on ${{\mathbb{R}} H^n} $ |
| title_full_unstemmed | Geodesic completeness of the left-invariant metrics on ${{\mathbb{R}} H^n} $ |
| title_short | Geodesic completeness of the left-invariant metrics on ${{\mathbb{R}} H^n} $ |
| title_sort | geodesic completeness of the left-invariant metrics on ${{\mathbb{r}} h^n} $ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/645 |
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