Nil Potent

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Since the time of Herman Weyl, geodesically equivalent metrics were ac-tively discussed in the realm of general theory. The context of general relativity

poses the following restrictions: the dimension is 4, the metrics are pseudo-Riemannian of Lorenz signature (, +, +, +) or (+, , , ), and sometimesthe metrics satisfy additional assumptions such that one or both metrics areRicci-flat (the vacuum Rij = 0), or Einstein (Rij =

R4

gij ), or, more generally,

satisfy the Einstein equation Rij R2 gij = Tij with physically interestingstress-energy tensor Tij . We refer to [8], [12], [26] for more details. We alsopoint out that geodesic equivalence of metrics is closely related to theory ofintegrable Hamiltonian systems (see for example [21].

Although, in general, for geodesically equivalent metrics g and g, their cor-responding Levi-Civita connections and may still be expected to differ, itturns out in many interesting situations they are necessarily equal. Intuitively,one might expect a link between projective relatedness and holonomy type and

if= is the result (

M, g) and (

M,g

) have the same holonomy type. Holon-omy theory can then describe precisely the (simple) relationship between g andg (and the signature ofg and g may differ) [10], [11].

Constant curvature metrics are not the only metrics that allow nontrivialgeodesical equivalence. For example, as it was shown by Dini [6] the followingtwo metrics on U2 R2 are geodesically equivalent

g= (X(x) Y(y)) (dx2 + dy2) and

1

Y(y) 1

X(x)

dx2

X(x)+

dy2

Y(y)

, (1)

whereXandYare arbitrary (smooth) functions of the indicated variables suchthat the formulas (1) correspond to metrics (i.e. 0= X= Y = 0 for all(x, y) U2). This example was generalized for all dimensions by Levi-Civita[18].

Anyhow we emphasize the existence of pro jective transformations is a strongercondition than the existence of geodesically equivalent metrics. We supportthese remark with the results as follow.

V.S. Matveev [25] (see also [22], [23], [24]) has proved that if a connectedLie group action on a complete Riemannian manifold preserves the geodesics(considered as unparameterized curves) then the metric has constant positivesectional curvature, or the group acts by affine transformations. He [26] alsogave an algorithm how to obtain a list of pairs of all possible geodesically equiv-alent 4-dimensional metrics g, g of Lorenz signature. This algorithm works inall dimensions, not only in dimension 4. G.S. Hall and D.P. Lonie [13] havealso constructed, using different approach, geodesically equivalent 4-dimensionalmetrics g, g of Lorenz signature. Anyhow, if one considers the same problem

of geodesically equivalent metrics within special classes of manifolds the sit-uation is different. As we mentioned below, manifolds of constant curvatureallow geodesically equivalent metrics (see [17], [1], [2], etc.). V. Kiosak andV.S. Matveev [14] have proved that every complete Einstein (Riemannian orpseudo-Riemannian) metric g of nonconstant curvature is geodesically rigid: ifany other complete metric g has the same (unparameterized) geodesics with g ,

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then the Levi-Civita connections ofg and g coincide. N.S. Sinjukov [28] provedthat all metrics given on a symmetric space, which are geodesically equivalent,

have the same Levi-Civita connection.All of these previously mentioned facts have motivated us to study geodesi-

cally equivalent metrics on 4-dimensional nilpotent groups as one of the nextsubclasses of homogeneous spaces. Moreover, let us point out among nilpotentgroups there are so called the Heisenberg type groups, which play an impor-tant role in geometric analysis, Lie groups and mathematical physics. It isshown that geometry of nilpotent groups vary if we replace Riemannian metricby pseudo-Riemannian one. Therefore, this is one reason more to investigatethem. We refer to [5], [9], [15], [16], and references therein for more details.

Our main result is the following theorem.MAIN THEOREM .............Our paper is organized as follows: .....

Preliminaries

In this Section we give known definitions and results which we use throughoutour paper.

Let n be a real nilpotent Lie algebra and let N be a Lie group having Liealgebra n and automorphism group Aut(N). If , is an inner product onn, we denote by (N, , ) the pseudo-Riemannian manifold N endowed withthe left invariant metric defined by , of arbitrary signature in general case.We are interested in this paper for Nof dimension 4 and Lorenz type metricof signature (+, +, +, ) or (, , , +). Let h3 and H3 be the Heisenberg Liealgebra and the Heisenberg Lie group. Let ={x1, x2, x3, x4} be a basis ofh

R with non-zero commutator

[x1, x2] = x3,

and let g4 be the Lie algebra satisfying

[x1, x2] = x3, [x1, x3] = x4. (2)

Let H3 Rand G4 be the simply connected Lie groups with Lie algebras h Rand g 4 respectively.

Magnin [20] proved that, up to isomorphism, there are only two non-Abeliannilpotent Lie algebras of dimension 4 : h R and g4.

The algebra h R is 2step nilpotent with two dimensional center Z(h R) = Rx4, x3 and one dimensional commutator subalgebra. Consider simplyconnected group N4 isomorphic to H3R associated with algebra h R. Wecan realize it as the real matrices

1 x z 0 00 1 y 0 00 0 1 0 00 0 0 1 w0 0 0 0 1

.

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So this group is diffeomorphic to R4. Instead of standard basis, for some com-putations we need also to take coordinate basis

v1 =

x, v2 =

y+ x

z, v3 =

z, v4 =

w

which realizes the structure equation in the given matrix representation.The algebra g4 is 3step nilpotent with one dimensional center Z(g4) =

Rx4and two dimensional commutator subalgebrag4 := [g4, g4] = Rx3, x4.

The Lie group corresponding to Lie algebrag4is realized by the real matrices

G4=

1 x x2

2 w

0 1 x z0 0 1 y

0 0 0 1

| x ,y ,z,w R

.

Left invariant vector fields

x,

y+ x

z+

x2

2

w,

z+ x

w,

w

in every point ofG4 form a basis ofG4.In [9] Guediri showed that on a 2-step nilpotent Lie group, all left-invariant

pseudoriemannian metrics are geodesically complete. In the same paper hefound an example of a non-complete left-invariant Lorentz metric on a 3-stepnilpotent Lie group. It turns to be exactly the group G4.

Let g be a Lie algebra. Denote by Aut(g) the group of automorphisms ofLie algebra g, that is

Aut(g) :={F :gg | F linear, [F x , F y] = F[x, y], x , yg}Denote bySL(g) set of non-equivalent Lorenzian inner products (i.e. of signa-ture (+, +, +, )) of algebra g. With basis of Lie algebra g fixed, the setSL(g)is identified with symmetric matrices Sof signature (n 1, 1), with n = dim g,modulo the following action of the automorphism group

SFTSF,F Aut(g).

Is is easy to find the group of automorphisms of algebras g4 and h R inthe basis .

Lemma 0.1 [16] The group Aut(g4) of automorphisms of Lie algebra g4, inbasis, consists of real matrices of form:

F=

a1 0 0 0a2 b2 0 0a3 b3 a1b2 0a4 b4 a1b3 a

21b2

,wherea1, b2= 0.

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Lemma 0.2 [16]

Aut(h R) = A 0B det A 0

AGL(2,R), = 0, R . (3)Moreover, if N is simply connected, then the exponential mapping exp :

nN is a diffeomorphism. We need not make distinction between automor-phisms ofn and those ofN [30, ?].

Geometries of these groups are different. We refer to [5], [16], [9], [27], etc.for more details.

We recall also basic formulas in the setting of geodesically equivalent metrics.We work in tensor notations with the background metricg. That means, we sumwith respect to repeating indexes, use g for raising and lowing indexes (unlesswe explicitly mention), and use the levi-Civita connection of g for covariantdifferentiation. As it was known already to Levi-Civita [18], two connections = ijk and =

ijk have the same unparameterized geodesics, if and only if

their difference is pure trace: there exists a (0, 1)-tensor such that

ijk = ijk +

ikj+

ij k . (4)

The reparameterization of the geodesics for and connected by (4) is doneaccording to the following rule: for a parameterized geodesic () of , thecurve ((t)) is a parametrized geodesic of , if and only if the parametertransformation(t) satisfies the following ODE:

=

1

2

d

dt

log

|d

dt|

. (5)

(We denote by the velocity vector ofwith respect to the parameter t, and

assume summation with respect to the repeating index.)If and related by (4) are Levi-Civita connections of metricsg and g, then

one can find explicitly (following Levi-Civita [18]) a function on the manifoldsuch that its differential , i coincides with the covectori: indeed, contracting(4) with respect toi andj , we obtaini =

i +(n+1)i. From the other side,

for the Levi-Civita connection of a metric g we have k = 12

log(| det(g)|)xk

.

Thus

i= 1

2(n + 1)

xilog

|det(g)det(g)

|

=,i (6)

for the function : M R given by

:= 1

2(n + 1)log

|det(g)det(g)

|

. (7)

In particular, the derivative ofi is symmetric, i.e., i,j =j,i.The formula (4) implies that two metrics g and gare geodesically equivalent

if and only if for certain i (which is, as we explained above, the differential of given by (7)) we have

gij,k 2gijk gikj gjk i = 0, (8)

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where comma denotes the covariant derivative with respect to the connection. Indeed, the left-hand side of this equation is the covariant derivative with

respect to , and vanishes if and only if is the Levi-Civita connection for g.The equations (8) can be linearized by the following substitution: consider

aij and i given byaij =e

2g gigj , (9)

i=e2ggi , (10)where g is the tensor dual to g : g

igj = ij . It is an easy exercise to

show that the following linear equations on the symmetric (0, 2)-tensoraij and(0, 1)-tensori are equivalent to (8)

aij,k = igjk + j gik . (11)

Remark. For dimension 2, the substitution (9,10) was already known to R.

Liouville [19] and Dini [6], see [3] for details and a conceptual explanation. Forarbitrary dimension, the substitution (9,10) and the equation (11) are due toSinjukov [29]. The background geometry is explained in [7].

Note that it is possible to find a function such that its differential isprecisely the (0, 1)-tensori: indeed, multiplying (11) byg

ij and summing withrespect to repeating indexes i, j we obtain (gij aij),k = 2k. Thus, i is thedifferential of the function

:=1

2ga . (12)

In particular, the covariant derivative ofi is symmetric: i,j = j,i.

1 Classification of non-equivalent Lorenzian in-

ner products

In this section we describe sets of non-equivalent Lorenzian inner products onh R and g4. To describe these sets we use geometrical approach, havingin mind the properties of the center of corresponding algebras and commutatorsubalgebrag4. Therefore these results are interesting independently of our mainidea of studying geodesically equivalent metrics.

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1.1 Case of algebra g4

Theorem 1.1 The setSL

(g4) of non-equivalent Lorenzian inner product ofalgebrag4 in basis{x1, x2, x3, x4} is represented by the following matrices

SA =

1 0 0 00 1 0 00 0 a b0 0 b c

, SA =

1 0 0 00 1 0 00 0 a b0 0 b c

,

S1 =

1 0 0 00 0 0 10 0 00 1 0 0

, S2 =

0 0 0 10 1 0 00 0 01 0 0 0

,

S3 =

0 0 1 00 1 0 01 0 0 00 0 0

, S4 =

1 0 0 00 0 1 00 1 0 00 0 0

,

with > 0, b 0. Matrix A =

a b

b c

satisfies det A > 0 in SA and

det A 0).Therefore, the orthogonal complement g4

is of signature (+, ) and g4 =

g4 g4. Let{e1, e2} be an orthonormal basis ofg4. . The parameter t Rof the hyperbolic rotation

f1 = cosh te1+ sinh te2, f2 = sinh te1+ cosh te2

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can be chosen in such way that the first coordinate off2 vanishes and thereforef2 is of the form (13). Such choice off1, f2 also determines basis

{f3, f4

}ofg4.

Therefore, in the new basis the matrix of the inner product, is of the formSA . The sign comes from the fact that the hyperbolic rotation cant changecharacter off2(i.e. signf2, f2). One can show that inner products representedby matrices SA are mutually non-equivalent under the action ofF Aut(g4).However, the automorphism F with nonzero parameters a1 =1, b2 = 1 canchange sign of parameter b in matrix A, so we have the condition b >0.

Case 2. g4 is of signature (+, ) (det A 0, N, X= 0.It is interesting that plane g4

is also degenerate and

g4 g4 = RN.

Therefore, sumg4

+g4 is three dimensional vector space, and doesnt span g4.

Case 3a) The center ofg4 is null.This means that central vector x4 satisfies 0 =x4, x4 = c. Therefore, we

get b = 0 and the matrix of restriction of inner product is A =

00 0

.

MatrixS reduces to

S=

p r e f

r q g h

e g 0f h 0 0

, >0.

Let f3 = x3, f4 = x4 and f1 = a1x1+ a2x2+ a3x3+ a4x4. The conditionthat f1g4, i.e.f1, f3= 0 =f1, f4is equivalent to

a1f+ a2h= 0, a1e + a2g+ a3= 0.Suppose that eh f g= 0, so this system has solution for a1, a2 with a3= 0 toassert that f1 is not collinear with f4 = x4. Let f2 = b2x2+ b3x3+ b4x4. Theconditionf1, f2= 0 reduces to

a4h(f g eh) + a3(egh + hr + f(g2 q)).

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for certain choice of parameterb4.It is easy to show that it is possible to choosevector f1 such that matrix of

,

in basis

{f1, f2, f3, f4

}is

S(,,t) =

0 0 t 00 0 0t 0 0 00 0 0

, , >0, t= 0.

By action ofAut(g4), similar to (15), it is possible to eliminate two parameters,and the matrix of inner product is reduced to S3 .

We skip similar calculation for the case g= 0, that results with inner productgiven by matrix S4 .

1.2 Case of algebra h

R

Group of automorphisms in basis {x1, x2, x3, x4} with structure equation [x1, x2] =x3, is easily computed:

Lemma 1.1

Aut(h R) = A 0

B det A

0

AGL(2,R), = 0, R

(16)

Theorem 1.2 All non-equivalent Lorenzian inner product of algebra h R inbasis{x1, x2, x3, x4} are represented by one of the following matrices

S = 1 0 0 0

0 1 0 00 0 00 0 0 1

, >0 S = 1 0 0 0

0 1 0 00 0 00 0 0 1

, >0

S10 =

1 0 0 00 1 0 00 0 0 10 0 1 0

S20 =

1 0 0 00 0 1 00 1 0 00 0 0 1

S30 =

1 0 0 00 0 0 10 0 1 00 1 0 0

.

Proof: We will be using similar approach as in case of algebra g4. Let S =(sij ), S = S

T, be the inner product matrix and let =x3, x4 denotes thecenter of algebra h R.Case 1. is of signature (+, +). Let us consider automorphism (16). It

is easy exercise to show that matrices A and B can be chosen in a way thatFTSF =S= diag(1, 1, d2, 1), = d2 >0.Case 2. is of signature (+, ). Therefore, the orthogonal complement isof signature (+, +). We can distinguish following cases:

Case 2a) Let|x3|2 =,|x4|2 =, , {1, 1}, =.

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For appropriate choice of matrix A in (16), we get FTSF=diag(1, 1, d2, ).Setting =

= 1, corresponding inner product is S+ and for =

=

1,

obtained inner product isS.

Case 2b) If|x3|2 =|x4|2 = 0,x3, x4 = 0, thenx3, x4 is positive-definiteand we can set x1, x2 x3, x4 : |x1|2 =|x2|2 = 1,x1, x2= 0.Case 3. Let be degenerate. We have only two possibilities: either x3 or x4are null.

Case 3a) If|x3|2 = 0, there exists v h R such thatx3, v = 0. If|v|2 = 0, we set x4 = 1x3,vv, otherwise x4 =

|v|22x3,v2 x3+

1x3,vv. Notice that

|x4|2 = 0,x3, f= 1.Plane spanned by x3, x4 is of signature (+, ) and x3, x4is positive-definite.Let us show that for some s

R, t

= 0 vector sx3+ tx4 belongs to

x3, x

4

:

x3, sx3+ tx4 = s2|x3|2 + stx3, x4= 0x4, sx3+ tx4 =

|v|22x3, v2 x3+

1

x3, vv,sx3+ tx4

= s|v|2

2x3, v2 |x3|2 t |v|

2

2x3, v2 x3, x4 + s

x3, vv, x3 + t

x3, vv, x4

= 1

x3, v(sv, x3 + tv, x4) .

Ifv, x4 = 0, set s = 0, t= 0, otherwise t = v,x3v,x4s, s= 0. Now vectorsx1, x

2 x3, x4 can be chosen such that

x4 = (sx3+ tx4), |x1|2 =|x4|2 = 1, x1, x4= 0.Let

x1 = a11x1 + a21x2 + a31x3 + a41x4x2 = a12x1 + a22x2 + a32x3 + a42x4x3 = x3x4 = a34x3 + a44x4, a44= 0

(17)

Settingd2 =a11a22 a12a21, we get:

Aut(h R)F =

a11a12

d 0 0

a21a22

d 0 0

a31a32

d d a34

a41a42

d 0 a44

, (18)

resultingFTSF=S20 .

Case 3b) Let|x4|2 = 0. Ifx3, x4 = 0, vector x4 = 2 x3|x3|2 1x3,x4x4 satisfiesrelations|x4|2 = 0,x3, x4= 1. Otherwise, set x4 = x4.

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Applying similar analysis as in previous case, we get vectors

x2 : |x2|2

= 0,x4, x2= 1, and x1 x4, x2 : x3, x1= 0,|x1|2

= 1.

In basis (17) automorphism Fis represented by matrix (18) and FTSF = S30 .

2 Curvature of g4 and h3 with respect to the

inner products

To calculate Levi-Civita connections we use Koszuls formula:

2X Y, Z = X.Y, Z Y.Z, X Z.X, Y+ [X, Y], Z [Y, Z], X + [Z, X], Y.

Since both vector fields and metric on the Lie group are left-invariant, thecomponents in the first row of Koszuls formula vanish.

We start from the commutators (2) and particular inner product, of thealgebra (Theorem 1.1) given by symmetric matrix S(it is similar in the case ofalgebrahR). Denotev =xixj , i , j = 1 . . . 4 the vector we want to calculate.From Koszuls formula we know

k :=v, xk= 12

([xi, xj ], xk [xj , xk], xi + [xk, xi], xj) .

Let Sbe matrix of, and [v] denotes column of coordinates of vector v , bothin basis{x1, x2, x3, x4}. Then

S[v] = (1, 2, 3, 4)T [v] = S(1, 2, 3, 4)T.

In this way we obtain vectors v =xixj , i.e. the Levi-Civita connection.For curvature tensor we use formula

R(X, Y)Z=X (Y Z) Y(X Z) [X,Y]Z.Note that the curvature operators R(xi, xj ), i , j = 1 . . . 4 belong to algebraso(S) preserving the inner product, i.t.

R(xi, xj)TS+ SR(xi, xj ) = 0.

Ricci curvature is defined as trace of the operator

(X, Y) = T r(ZR(Z, X)Y),with components ij = (xi, xj ). Scalar curvature is the trace of the Riccioperator

Scal=

i

ii =

i,j

gijji ,

where we have used the inverse{gij} = S1 of the metric to raise the index(i.e. to obtain Ricci operator).

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2.1 Case of algebra g4

Case 1) Three families of inner products SA and SA can be written using thematrix

S=

1 0 0 00 2 0 00 0 a b0 0 b c

,

with1, 2 {1, 1} and sign(detA) =12. Denote d = ac b2 = det A.Nonzero covariant derivatives are

x1x2 = 1

2x3 =x2x1,

x1x3 = a2

2 x2 bc

2dx3+

ac

2dx4 =x3x1+ x4,

x1x4 = b22 x2 c2

2dx3+ bc

2dx4 =x4x1,

x2x3 = a1

2 x1=x3x2,

x2x4 = b1

2 x1 =x4x2,

x3x3 = b1x1,x3x4 =

c1

2 x1 =x4x3.

All curvature operators R(xi, xj ) are nonzero. For example we obtain:

R(x2, x4)x1 = 0, R(x2, x4)x2 =b14

x3, R(x2, x4)x3 = b1

4d(ad2 x2+bc x3ac x4),

R(x2, x4)x4 = b1

4d(bd2 x2+ c

2 x3 bc x4).Non-zero components of Ricci tensor are

11 =c2 + 2ad

2d , 22 =1 a

2, 23 =1 b

2,

33 = 1

2d(2a

2d cd + cb2), 34 = 1b2d

(c2 + 2ad), 44 = 1

2d(2b

2d + c3).

It is easy to check that Einstein condition implies b = 0, c = 0, what isimpossible. The scalar curvature is

Scal = 12d

(c2 + 2ad).

It is clear that scalar curvature can take any real value, including zero, for somechoice of parameters a,b,c.

Case 2)Matrix of inner product is of the form S1 , >0.

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Nonzero covariant derivatives are

x1x2 = 12 x3=x2x1+ x3,

x1x3 = 1

2 x4 =x3x1+ x4,

x2x3 = + 1

2 x1=x3x2.

Nonzero components of curvature tensor are:

R(x1, x2)x1 = (3 1)( + 1)

4 x4,

R(x1, x2)x2 = (3 1)( + 1)4

x1,

R(x2, x3)x2 = ( + 1)2

4 x3,

R(x2, x3)x3 = ( + 1)2

4 x4.

The only nontrivial component of Ricci tensor is 22 =212 . Since >0, wehave that metric is Ricci flat for = 1.The group is not Einstein and the scalarcurvature equals 0.

Case 3)Matrix of inner product is of the form S2 , >0.Nonzero covariant derivatives are

x1x1 = 1 x3,

x1x2 = 1

2x3 =x2x1,

x1x3 =

2x2+ x4 =x3x1+ x4,

x2x3 =

2x4 =x3x2


R(x1, x2)x1 = 3

4 x2,

R(x1, x2)x2 = 3

4 x4,

R(x1, x3)x1 = 4

x3,

R(x1, x3)x3 = 2

4 x4.

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The only nonzero component of Ricci tensor is 11 =2 , and the group is notEinstein. The scalar curvature equals 0.


x1x1 = x2,x1x2 = x3,x1x3 =

1

2x4=x3x1,

x1x4 =

2x1 =x4x1,

x2x4 =

2x3 =

x4x2.


R(x1, x2)x1 = 1

2x4, R(x1, x2)x4 =

2x3,

R(x1, x3)x1 = 3

4 x1, R(x1, x3)x3 =3

4 x3,

R(x1, x4)x1 =

2x2, R(x1, x4)x2 =

2x3,

R(x1, x4)x3 =

4x4, R(x1, x4)x4 =

2

4 x1.

Nonzero component of Ricci tensor are

13 =

2, 44 =

2

2 .

and the group is not Einstein. The scalar curvature can take any value

Scal=

2.


x2x1 =

x3,

x1x3 = 12 x4=x3x1,

x1x4 =

2x2 =x4x1,

x2x4 =

2x1 =x4x2.

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R(x1, x3)x1 = 34

x2, R(x1, x3)x3 =34

x1,

R(x2, x3)x2 = 1

2x4, R(x2, x3)x4 =

2x3,

R(x2, x4)x2 =

2x2, R(x2, x4)x3 =

4x3,

R(x3, x4)x3 = 4

x4, R(x3, x4)x4 = 2

4 x2.

Nonzero component of Ricci tensor are

24 =

2, 33 =

2.

and the group is not Einstein. The scalar curvature equals 0.

2.2 Case of algebra h RCase 1) If we take nontrivial inner products

x1, x1= 1, x2, x2=1, x3, x3= >0, x4, x4= 1,the only nontrivial covariant derivatives are

x1x2 = x2x1 = 12x3x1x3 = x3x1 = 2 x2x2x3 = x3x2 = 2 x1.Now it is easy to compute curvature tensor

R(x1, x2)x1 = 34 x2 R(x1, x3)x1 = 4 x3 R(x2, x3)x2 = 4 x3R(x1, x2)x2 = 34 x1 R(x1, x3)x3 =

2

4x1 R(x2, x3)x3 =

2

4x2

and non-zero components of Ricci tensor

(x1, x1) =(x2, x2) = 2

, (x3, x3) =2

2 .

The scalar curvature is positive

Scal=

2.

Case 2)Similarly, for inner products S, >0 andS+,


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(x1, x1) = (x2, x2) =2

, (x3, x3) = 2

2 .

The scalar curvature can take any value

Scal=2

.

Case 3) Performing similar analysis, we can show that for inner product S01the only non-zero components of curvature tensor are

R(x1, x4)x1 = 14x3R(x2, x4)x2 = 14x3R(x1, x4)x4 =

14

x1R(x2, x4)x4 =

14x2.

Ricci tensor has only one nontrivial component (x4, x4) = 12 . The scalar cur-

vature equals 0.

Case 4) Observe that for inner product S20 all covariant derivatives but theones deriving from the structure equation vanish, so we have that

R(x1, x2) = R(x2, x3) = R(x3, x1) = 0

and the corresponding metric is flat.

Case 5)In partly null and partly orthonormal basis non-trivial connections are

x1x2 = x2x1 = 12x3x1x3 = x3x1 = 12x4x2x3 = x3x2 = 12x1.

Space H3R is not flat; in fact, a direct computation shows thatR(x1, x2)x2, x1 = 34x1, x1 = 34R(x3, x2)x2, x3 = 14x3, x3 = 14 .

Non-vanishing component of Ricci tensor is(x2, x2) =12 and scalar curvatureis 0.

3 Left-invariant metrices

Let us consider left-invariant basis:

x,

y+ x

z+

x2

2

w,

z+ x

w,

w. (19)

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Theorem 3.1 Each left-invariant Lorentzian metric on the group G4 is iso-metric to one of the following metrices

gA = dx2 dy2 + a(xdy dz)2 b(xdy dz)(2dw 2xdz+ x2dy) +c

4(2dw+ x(xdy 2dz))2

gA = dx2 + dy2 + a(xdy dz)2 b(xdy dz)(2dw 2xdz+ x2dy) + c

4(2dw+ x(xdy 2dz))2

g1 = dx2 + 2dwdy+ xdy(xdy 2dz) + (xdy dz)2

g2 = 2dwdx + dy2 + xdx(xdy 2dz) + (xdy dz)2

g3 = dy2 2dx(xdy dz) +

4(2dw+ x(xdy 2dz))2

g4 = dx2 2dy(xdy dz) +

4(2dw+ x(xdy 2dz))2.

All of them are non-isometric.

We can take basis ofN4

x,

y+ x

z,

z,

w (20)

that is left-invariant.

Theorem 3.2 Each left-invariant Lorentzian metric on the group N4 is iso-metric to one of the following metrices

g = dx2 dy2 + (xdy dz)2 + dw2

g = dx2 + dy2 (xdy dz)2 dw2

g10 = dx2 + dy2

2xdydw+ 2dzdw

g20 = dx2 2xdy2 + dw2 + 2dydzg30 = dx

2 + 2dydw+ (xdy dz)2

and all of them are non-isometric.

4 Projective equivalence

Consider the projective Weyl tensor

Wijkl := Rijkl

1

3

il Rjk ikRjl

.

In order to find he projective class of our 4-dimensional metrics, we use the

formula2ga(iWj)akl = gabW(iab[l

j)k]

, (21)

proposed by Matveev in [26], where the brackets [ ] denote the skew-symmetrizationwithout division, and the brackets ( ) denote the symmetrization without di-vision.

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Case of group G4

original metric equivalent metric

gA =

1 0 0 00 2 0 00 0 p q0 0 q r

gA =

c1p

1 0 0 0

0 c1p

2 0 0

0 0 c1 00 0 0 rc1

p

= c1pg, q= 0, p= 0c1 R \ {0}

1, 2 {1, 1}

gA =

c1qd

1 0 0 00 c1q

d 2 0 0

0 0 c1pqd

c1q2

d

0 0 c1q2

dc1qr

d

= c1qd g, q= 0, d= q2 pr= 0c1 R \ {0}

g1 = 1 0 0 00 0 0 10 0 00 1 0 0

g1 = c1 0 0 00

c2

1c2

0 c10 0 c1 0

0 c1 0 0

, c1, c2 R, c1= 0

g2 =

0 0 0 10 1 0 00 0 01 0 0 0

g2 =

c21c2 0 0 c10 c1 0 00 0 c1 0

c1 0 0 0

, c1, c2 R, c1= 0

g3 =

0 0 1 00 1 0 01 0 0 00 0 0

g3 =

0 0 c1 00 c1 0 0

c1 0 0 00 0 0 c1

=c1g, c1 R \ {0}

g4 =

1 0 0 00 0 1 0

0 1 0 00 0 0

g

4 =

c1 0 0 00 0 c1 0

0 c1 0 00 0 0 c1

=c1g, c1 R \ {0}

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Case of group N4

original metric equivalent metric

g =

1 0 0 00 1 0 00 0 00 0 0 1

g=

c1 0 0 00 c1 0 00 0 c1 00 0 0 c2

, c1, c2 R, c1= 0

g =

1 0 0 00 1 0 00 0 00 0 0 1

g =

c1 0 0 00 c1 0 00 0 c1 00 0 0 c2

, c1, c2 R, c1= 0

g10 =

1 0 0 00 1 0 00 0 0 1

0 0 1 0

g10 =

c1 0 0 00 c1 0 00 0 0 c1

0 0 c1 c21c2

, c1, c2 R, c1= 0

g30 =

1 0 0 00 0 0 10 0 1 00 1 0 0

g30 =

c1 0 0 00 c21c2 0 c10 0 c1 00 c1 0 0

, c1, c2 R, c1= 0

5 Isometry groups

We have previously denote by Aut(N) the automorphism group ofNand let usdenote by I(N) the isometry group of N. Set O(N) = Aut(N) I(N) and letO(N) be the subgroup ofI(N) that fixes the identity element ofN .Cordero and Parker proved in [4] the following result for 2-step nilpotent groups:

Theorem 5.1

a) If the center of N is degenerate. Then I(N)O(N) N, where N actsby left translations.

b) If the center of N is non-degenerate. ThenO(N) O(N) and I(N)O(N) N.

c) If N is flat. ThenO(N) O(p, q), where O(p, q) denotes the pseudo-orthogonal Lie group and(p, q)is the signature of the pseudo-Riemannianmetric.

5.1 Case of group G4

Let us consider the 3-step nilpotent groupG4.

Lemma 5.1 If g denotes one of left invariant metrics gA, gA , g

1 defined in

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[27] N. Rahmani, S. Rahmani,Lorentzian Geometry of the Heisenberg Group,Geom. Dedic. 118 (2006), 133-140, DOI 10.107/s10711-005-9030-3

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