# Domain walls of N=2 supergravity in five dimensions from hypermultiplet moduli spaces

###### Abstract:

We study domain wall solutions in d=5, N=2 supergravity coupled to a single hypermultiplet whose moduli space is described by certain inhomogeneous, toric ESD manifolds constructed recently by Calderbank and Singer. Upon gauging a generic isometry of these spaces, we obtain an infinite family of models whose “superpotential” admits an arbitrary number of isolated critical points. By investigating the associated supersymmetric flows, we prove the existence of domain walls of Randall-Sundrum type for each member of our family, and find chains of domain walls interpolating between various backgrounds. Our models are described by a discrete infinity of smooth and complete one-hypermultiplet moduli spaces, which live on an open subset of the minimal resolution of certain cyclic quotient singularities. These spaces generalize the Pedersen metrics considered recently by Behrndt and Dall’ Agata.

^{†}

^{†}preprint: YITP-SB-02-45

###### Contents

## 1 Introduction

Five dimensional gauged supergravity has acquired some
phenomenological interest due to several recent developments.
The work of [1] showed that the reduction of
Horava-Witten theory [2] to five dimensions is a gauged minimal
supergravity admitting a BPS saturated
domain wall solution which can be identified with the four-dimensional
space-time of a strongly-coupled heterotic compactification [3].
Another direction is provided by the AdS/CFT correspondence
[4]. In this framework the domain walls of
gauged supergravity have a natural interpretation as renormalization
group flows in the corresponding field theory.
When an embedding of supergravity into the theory is
known, the associated domain walls of the theory aquire an
RG flow interpretation^{1}^{1}1It has become customary to use
RG flow terminology even when such an embedding is not known, and we shall
do so in what follows..
Yet another development is the proposal
of [5] for an alternative to compactification. This scenario requires a
domain wall interpolating between two solutions (of equal
vacuum energy density) associated
with IR points (critical points for the “superpotential”
where the warp factor is exponentially small).
Despite intense interest in the subject, there
has been limited progress in finding explicit supergravity realizations
of such scenarios.

In this regard several no-go theorems were proposed [6, 7, 8, 9], which state that, under certain assumptions, there are no supersymmetric domain wall solutions connecting IR critical points of the supergravity potential. As it turns out, the relevant assumptions can be violated once one considers coupling to hypermultiplets. In particular, the recent work of [10] provides a counterexample obtained by coupling the supergravity multiplet to a single hypermultiplet described by a certain non-homogeneous quaternion-Kahler space; in this model, the no-go theorems of [6, 8, 9] do not apply. This underscores the importance of reconsidering the problem in the general context of inhomogeneous hypermultiplet moduli spaces.

As a general rule, however, one knows quite a bit about flows on the
vector/tensor multiplet moduli space, but rather little about their
hypermultiplet counterpart. The difficulty in the latter case
consists mainly in understanding the associated geometry. It is well-known
that the hypermultiplet moduli space must be a quaternion-Kahler space
of negative scalar curvature. To trust the supergravity
approximation, one must restrict to smooth quaternion-Kahler
spaces^{2}^{2}2In principle, one may allow for curvature
singularities in the classical hypermultiplet moduli space.
However, one expects such singularities to be removed by quantum
effects, for example if the model under consideration can be realized
in string/M-theory..
Even restricting to one hypermultiplet (the focus of the present paper),
very little is known explicitly for the generic case. In this situation,
the quaternion-Kahler condition is equivalent to the requirement that
is Einstein-selfdual (ESD). The simplest negative curvature examples
are provided by the homogeneous spaces (the moduli
space of the universal hypermultiplet, i.e. the Bergman metric)
and (the Euclidean version of , also
known as the hyperbolic space or the hyperbolic metric on the
open four-ball). Another class of examples is provided by
cohomogeneity one -invariant complete ESD metrics, which were
classified in [17]. A distinguished subclass of the latter is
provided by those metrics which admit an isometric action.
These are the Pedersen metrics on the open four-ball [26]
and their analytic continuations [26, 17]. As it turns
out, these are the metrics relevant for
the counter-example of [10]^{3}^{3}3The authors of [10] use a
parameterization due to [28], which is quite different from that
of [11] and [26], and somewhat cumbersome for our
purpose. The relation between their coordinates and those of
[26] is described in Appendix A..

What will allow us to make progress is the recent work of
Calderbank and Pedersen [11], which gave an explicit description
of the most general ESD space admitting two commuting and linearly
independent Killing vector fields. Through an elegant chain of
arguments, they showed that such spaces are described by a single
function of two variables, which is constrained to obey a linear PDE (namely, must be an eigenfunction of the
two-dimensional hyperbolic Laplacian with eigenvalue ). An
immediate consequence of this linear description is that one can
obtain new solutions (at least locally) by superposing various
eigenfunctions --- a situation which is quite unexpected at first
sight.^{4}^{4}4Note that an abstract classification of quaternion-Kahler spaces with quaternionic abelian isometries in terms of a single function was infered in [12], based on the relation to hyperkahler cones with abelian triholomorphic isometries. However in [12] the relation between the quaternion-Kahler metric and the single function characterizing it is rather implicit and hence more of conceptual than practical significance.

For the case of positive scalar curvature, the work of [11] has another application: it leads to an elegant description of ESD metrics on certain compact toric orbifolds which include and generalize the models studied a while ago by Galicki and Lawson [18]. As explained in [20] and [21], this can be combined with the construction of [19] and [23] in order to produce a large class of conical metrics, which lead to interesting M-theory backgrounds which produce chiral field theories in four dimensions [22]. The main simplification for the positive curvature case is due to Myers’s theorem, which forces such spaces to be compact (if complete); this makes them amenable to (hyperkahler) toric geometry techniques upon invoking the associated hyperkahler cone/Swann bundle. In particular, this allows one to extract global information by simple computations in integral linear algebra.

When studying hypermultiplets, the ESD
space of interest has negative scalar curvature and
the isometries of may fail to be
compact. Therefore, toric geometry techniques do not always apply.
This reflects the well-known observation that the global
study of Einstein manifolds of negative scalar curvature is
considerably more involved than the positive curvature case. In
particular, it is not trivial to find functions for which the
metric of [11] is smooth and complete as required by supergravity
applications. A class of such solutions was recently given
by Calderbank and Singer [24], and in this paper we shall restrict to
negative curvature models of that type. The metrics of
[24] are smooth and complete, and live on an open subset of
a toric resolution of an Abelian quotient singularity ;
the set contains all irreducible components of the exceptional
divisor. To ensure negative scalar curvature, one must require
^{5}^{5}5Compare this with the Gorenstein case
( surface singularities), which leads to the well-known
hyperkahler metrics of Eguchi–Hanson and Gibbons–Hawking
[25].. These models
admit two Killing vectors with compact orbits, and thus they are
invariant under a action. As pointed out in [11, 24],
such models are a generalization of the Pedersen–LeBrun metrics
[26, 27];
the latter arise as the particular case when is the minimal resolution of with
the symmetric action . As mentioned above, the Pedersen metrics
are invariant with respect to a larger action
and fit into the cohomogeneity one classification of invariant
ESD metrics given by Hitchin [17].

The main advantage of the metrics of [24] is that the underlying manifold admits a toric description, even though the metrics themselves have negative scalar curvature. Indeed, the resolution is a (noncompact) toric variety in two complex dimensions, whose combinatorial description is a classical result [29, 31]. In particular, the orbits of the isometric action can be described by standard methods of toric geometry [30, 31, 32, 33]. When considering flows on such spaces, this allows for easy identification of the critical points of the relevant superpotential since, in the absence of vector/tensor multiplets, the latter are the fixed points of the U(1) isometry used to gauge the supergravity action [14, 15].

In fact, the basic picture can be explained quite easily in
non-technical language. Recall that the toric variety can be
presented as a - fibration over its Delzant polytope
^{6}^{6}6This fibration arises by considering the moment
map of the action with respect to the
toric Kahler metric of . The Delzant polytope
is the image . Note that the toric Kahler
metric of differs from its ESD metric.. In our
case, the latter is a noncompact planar polytope and general results
[31, 33] show that the fiber of collapses to a
point at its vertices and to a circle above its edges. The
fibrations above the finite edges (whose circle fibers collapse to
points at the vertices) give a collection of smooth spheres
which are holomorphically embedded in — one obtains a copy of
for every finite edge of . A generic isometry fixes only the
points of sitting above the vertices of . As one
can obtain an arbitrary number of points by taking ( is
the resolution of ) to be large, one can produce
models with an arbitrarily large number of isolated critical points of
the superpotential. This should be contrasted with the Pedersen
metrics considered in [10], which lead to at most two isolated
critical points. This observation will allow us to build chains of
flows connecting the critical points, and therefore domain wall
solutions which interpolate between the associated
backgrounds.

For a general choice of gauged isometry, it turns out that at most one such flow is of Randall-Sundrum type (i.e it connects two IR critical points). Among the rest, there are domain wall solutions which interpolate between a UV and an IR critical point. Some of these may describe RG flows of appropriate dual field theories due to the following chain of arguments. It is believed that 5-dimensional gauged supergravity [35, 36] is a consistent truncation of the 10-dimensional IIB supergravity on (some evidence for this was presented in [37, 38]). This means that every solution of the former is also a solution of the latter. Although there is no proof at present, this is strongly supported by similarity with two other cases: 4-dimensional gauged supergravity, which is known [39] to be a consistent truncation of 11-dimensional supergravity on , and 7-dimensional gauged supergravity, which was shown recently [40] to be a consistent truncation of 11-dimensional supergravity on . Additional, indirect evidence for the consistency of the truncation of IIB SUGRA on to , SUGRA is provided by the numerous studies in (for example [41, 42]), where that consistency is assumed and domain walls of the supergravity theory are interpreted as RG flows of the corresponding dual field theory. Various quantities, calculated both from the gravity side and from the field theory side, have been successfully matched [41, 42]. For one of these solutions — a domain wall in 5d supergravity, which describes an RG flow from to super Yang-Mills driven by the addition of a mass term for one of the three adjoint chiral superfields [41] — it was found in [14] that it can be embedded in 5d gauged supergravity. In particular this means that the theory is at least in that case a consistent truncation of the 10d IIB theory. This may be by chance, but it may also be that many more domain walls of 5d SUGRA can be embedded in the 10d theory and hence can have via the correspondence an interpretation as a RG flow of an appropriate field theory. It would be interesting to explore whether some of our UV-IR flows have such interpretations.

The present paper is organized as follows. In Section 2 we recall the necessary ingredients of 5-dimensional gauged supergravity, and extract the superpotential and flow equations relevant for coupling the gravity multiplet to a single hypermultiplet described by the metric of [11]. Section 3 describes the subclass of metrics constructed in [24]. In Section 4 we study the general properties of BPS flows for such models. In particular, we give the general flow solutions connecting our critical points and describe the conditions under which such a solution has Randall-Sundrum type (i.e. is an IR-IR flow). Section 5 illustrates this discussion with a few explicit examples, which include and generalize the solution of [10]. Appendix A gives the coordinate transformation relating the Pedersen metrics to the models discussed in [10].

## 2 Flow equations on toric one-hypermultiplet moduli spaces

Consider coupling a single hypermultiplet to the supergravity
multiplet in five dimensions. As the theory contains only one gauge
field (namely the graviphoton), one can gauge at most one isometry of
the hypermultiplet moduli space. The general Lagrangian of gauged
minimal supergravity in five dimensions was derived in
[13]^{7}^{7}7Minimal means as in 5d there is no
gauged supergravity theory.. When no vector/tensor multiplets are
present^{8}^{8}8When including vector/tensor multiplets, the potential can
not always be written in this form. See [14] for details., the
scalar potential induced by the gauging takes the form [14]:

(1) |

where is the hypermultiplet metric and the ”superpotential” is given by:

(2) |

Here is the triplet of prepotentials related to the Killing vector of the gauged isometry:

(3) |

where is the triplet of Sp(1) curvatures and is the full covariant derivative (including the Levi-Civita, Sp() and Sp(1) connections for the case of a -dimensional quaternionic space). The factor of is a result of a different normalization w.r.t. [14, 10] as will be explained below.

Note that we work with the convention [14] that is always non-negative (i.e. we choose the non-negative square root in (2)). This implies that will fail to be differentiable at a zero of where the directional derivatives do not all vanish. At such a noncritical zero, the function is differentiable, and its Hessian is positive semidefinite [15].

In our example, we take and let the hypermultiplet moduli space be described by a invariant ESD metric of negative scalar curvature. As shown in [11], the most general invariant ESD metric has the form:

(4) |

where , and the function satisfies the equation:

(5) |

Note that we use the notation , etc. for the partial derivatives of .

In equation (4), one takes and , while are coordinates of periodicity . The metric is well-defined for and . It has positive scalar curvature in the regions where and negative scalar curvature for . One can easily check that (4) is normalized so that in the latter case the scalar curvature is as is usual for supergravity applications.

Condition (5) says that is an eigenfunction (with eigenvalue ) of the hyperbolic Laplacian . This is the Laplacian of the standard metric:

(6) |

on the hyperbolic plane with coordinates and . Note that we use the upper half plane model.

It is easy to check that they satisfy:

(8) |

unlike (2.11) of [14]. Hence these curvatures are normalized to

(9) |

where are the three complex structures, and not to as in [14, 10]. With this normalization the covariant derivative takes the form

(10) |

on a quantity having only an Sp(1) index. Note the slight difference w.r.t. (2.12) of [14]. In (10) is the triplet of Sp(1) connections for the curvature (2) [11]:

(11) |

Using (8) one can solve (3) for the Killing vector:

(12) |

Again the numerical factor differs slightly from [14, 10] due to the different normalization of the Sp(1) curvatures and connections. Equation (3) can also be solved for the prepotentials by using the fact that they are eigenfunctions of the Laplacian [16]:

(13) |

(for a -dimensional quaternion-Kahler space). Again this differs by a factor of w.r.t. the footnote after (2.15) in [14]. One obtains:

(14) |

where the second equality results from the covariant constancy and antisymmetry in and of the Sp(1) curvature. A domain wall of gauged supergravity,

(15) |

which preserves supersymmetry is given by the solution of the following system [14]:

(16) |

where are the hypermultiplet scalars. The signs in these equations must be chosen consistently (i.e. one must use the minus sign in the second equation if one chooses the plus sign in the first). In order to insure continuity of the derivative of a flow which passes through a noncritical zero of , one must switch the sign in the first equation when the flow meets such a point. Accordingly, the sign in the second equation must also be switched there.

Let us gauge the isometry of the metric (4) along the Killing vector:

(17) |

Note that we normalize such that the coefficient of its component equals one.

#### Observation

Equation (3) fixes the prepotentials (and thus the superpotential (2)) in terms of a specific choice for the Killing vector . In particular, rescaling leads to a rescaling of , which can be absorbed by a rescaling of in equations (2). Since the first of these equations determines only up to a constant factor, this rescaling of can be further absorbed into a constant rescaling of the metric (15), upon choosing an appropriate integration constant for . In particular, the normalization in (17) amounts to a particular choice of scale for or, equivalently, a choice of scale for the metric (15).

Equation (14) implies:

(18) |

where we took (5) into account. Using (12), one can check that these prepotentials give (17). Now (2) gives:

(19) |

Because is only a function of and and the metric (4) (and hence its inverse) is diagonal in , the potential (1) and the ”flow equations” (2) acquire a particularly simple form:

(20) |

and:

(21) |

Hence the isometry of the one-hypermultiplet moduli space allows us to reduce the four-dimensional flow equations to a two-dimensional problem. The last two relations describe the gradient flow of with respect to the metric on the upper half plane (notice that this is conformal to the hyperbolic metric).

## 3 Calderbank-Singer spaces

### 3.1 Minimal resolutions of cyclic quotient singularities

Consider a cyclic singularity , where the generator of acts through:

(22) |

We assume that the integers satisfy ^{9}^{9}9 This
action embeds diagonally in . It embeds in if and only
if , when the singularity is called Gorenstein and has trivial dualizing sheaf; in that case, it is
simply an surface singularity. In this paper, we are
emphatically not interested in the Gorenstein case. We note that
non-Gorenstein cyclic singularities arise naturally in the study of
normal complex surfaces–this generalizes the better known case of
singularities, which give local descriptions for the
singularities of . The minimal resolution of an
singularity has trivial first Chern class and carries the
multi-Eguchi-Hanson metric, which is hyperkahler. As shown in
[24], such resolutions never carry a toric SDE metric of
negative scalar curvature.. We consider the minimal resolution
of this singularity, which is a smooth algebraic surface
birational with and containing no curves. If
denote the irreducible components of its exceptional
divisor, then it is a classical fact [29] that the intersection
matrix of these components has the form:

(23) |

where the diagonal entries are integers satisfying . The adjunction formula shows that , with if and only if all and iff all ; the latter case corresponds to (the Gorenstein singularity). For what follows, we shall consider exclusively the case .

It is well known that both and its minimal resolution are
toric varieties (see, for example, [31]). As explained in
[31], the toric description of can be extracted with the
help of continued fractions. Indeed the integers and are given by the minus^{10}^{10}10This differs from the more
common ‘plus’ continued fractions. By definition of the expansion
(24), the integers are required to satisfy .
continued fraction expansion:

(24) |

which we shall denote by for simplicity. The toric
data of can be determined as follows [31]^{11}^{11}11Our
presentation differs from that of [31] in a few trivial
ways. First, reference [31] uses a different description of the
cyclic action, which amounts to the redefinitions and
. It also writes our second order recursion as a
first order recursion for two vectors.. Consider a basis of
the two-dimensional lattice , and define vectors by the two-step recursion:

(25) |

with the initial conditions and . Then and are the toric generators of the minimal resolution , while and are the toric generators of the singularity . The latter generate a (strongly convex) cone , subdivided by the vectors which lie in its interior. In fact, these vectors coincide with the vertices of the convex polytope defined by the convex hull of the intersection of with the lattice. Following [24], we choose and (always possible via a modular transformation), which gives , and . Upon writing , relation (25) becomes:

(26) |

with the initial conditions and . These can be recognized as the standard recursion relations for the numerator and denominator of the partial quotients () of the continued fraction (24). We remind the reader that the solutions of this recursion have the following properties (all of which can be checked by direct computation or induction):

(a) and

(b) .

(c) for and for .

(d) If all are strictly greater that two, then for all and for all .

The first part of (c) says that the area of the triangle determined by vectors and equals — this is the condition that the subdivision of the cone resolves the singularities of .

The situation for the vectors is illustrated in figure 1. It shows the case and , which will be discussed in more detail in Subsection 5.3.

### 3.2 The Calderbank-Singer metrics

In [24], Calderbank and Singer construct toric ESD metrics of
negative scalar curvature on the minimal^{12}^{12}12In fact, the
construction of [24] applies to a more general class of toric
resolutions of , which are not necessarily minimal. In this
paper, we shall consider their construction only for the case of
minimal resolutions. resolutions of cyclic singularities with
negative first Chern class. These metrics are invariant with respect
to the natural action on induced by its structure of toric
variety. In view of the results of [11], they must have the
general form (4) with , where the angular coordinates parameterize
the fibers of . This is achieved by choosing to be a
superposition of elementary eigenfunctions of the type:

(27) |

which are easily seen to satisfy . More precisely, one must choose the linear combination:

(28) |

where:

(29) |

and we defined and (addition of and thus of amounts to taking the one-point compactification of ). Since we assume , we have for all and thus . The combination (28) is fixed (up to a constant scale factor) by the requirement that the metric (4) extends smoothly to the singular fibers of . When considering the metric (4), one identifies topologically the boundary of the Delzant polytope with the boundary of the upper half plane model of the hyperbolic plane. Then the vertices of the Delzant polytope are mapped to the points , and its edges correspond to the intervals determined by these vertices. It is easy to check [24] that , so that the edges of correspond to the intervals sitting at .

Expression (4) determines an ESD metric (of negative scalar curvature) on the space , where is the conformal compactification of the hyperbolic plane (obtained by adding the point at infinity). However, this metric is ill-defined along the set given by the equation . As explained in [24], this locus is a smooth simple curve which intersects the boundary of in the points and (see figure 2). This curve separates into connected components (defined by the condition ) and (defined by , which pull back to two disjoint open subsets and of , separated by a region defined as the pull-back of . The piece contains the points , while contains the point . The set is a compact fibration over , which coincides topologically with the Lens space . It is a conformal infinity for each of the two ESD metrics determined by (4) on the open sets and [11]. Note that contains the exceptional divisor of the resolution, and that the ESD metric induced on is smooth and complete. The metric induced on is a complete orbifold metric, with the orbifold point given by the point at infinity of , which we denote by (this points sits above ). Since we are interested in smooth and complete metrics, we shall concentrate on the region (see figure 3).

### 3.3 Fixed points of isometries

The fibration of over its Delzant polytope translates into a fibration over . Since the points correspond to the vertices of , the fibers collapse to points above and to circles above the intervals sitting at . In expression (4), one uses coordinates along the fibers such that the vectors correspond to the canonical basis of the lattice . Hence the generator (17) corresponds to the two-vector . This isometry fixes the sphere lying above precisely when is orthogonal to the generator , i.e. when . Note that all isometries (17) fix the points of lying above .

## 4 Supersymmetric flows on Calderbank-Singer spaces

### 4.1 Critical points of the superpotential

When the supergravity multiplet is coupled only to hypermultiplets but not to vector/tensor multiplets, it was shown in [14] and [15] that the critical points of the superpotential (19) coincide with the fixed points of the associated isometry (17). In view of the discussion above, we find that an isometry of with different from fixes exactly the points ; thus a generic isometry has critical points. In the non-generic cases , the isometry fixes together with the entire sphere .

### 4.2 Asymptotic form of the flow equations and divisorial flows

The superpotential (19) can be written:

(30) |

where we used on the domain of interest . Let us write the flow equations (2) as:

(31) |

where:

(32) |

It is not very hard to check the following asymptotics for as :

(33) |

To arrive at these expressions, we defined:

(34) |

In particular, one has , so that the gradient lines of become orthogonal to the real axis for . Thus one can find a flow (integral curve) along this axis by setting consistently in equations (4.2). In this case, the second equation in (4.2) is trivially satisfied and the system reduces to:

(35) |

where:

(36) |

Up to the factor , the second equation in (4.2) describes the one-dimensional gradient flow of the function:

(37) |

with respect to the limiting metric:

(38) |

This induced metric blows up on the interval precisely at
the points (), but this is a coordinate
singularity. Note that (38) is
continuous on . It is also
clear^{13}^{13}13This can be checked directly by noticing that
blows up like for
. Hence the distance
stays finite. It also
follows from the fact that the metric of [24] is adapted to the
toric fibration , which restricts to
the two-sphere over each interval
. that the length of each interval is
finite with respect to this metric for . The intervals
and have infinite length since they
bound the conformal infinity .

Since the region of sitting above each interval ) is the 2-sphere (a component of the exceptional divisor), it is clear that flows of type (4.2) lift to flows in which are entirely contained inside some . The intersection matrix (23) shows that the dual graph of is a chain, so touches only and for . Let be a coordinate along the uncollapsed circle above . Since and the metric (4) are independent of the fiber coordinates, the flow equations (2) require (this can also be seen from equations (2), since are certain linear combinations of the angular coordinates ). Hence the flow proceeds along the sphere at some fixed angular value (figure 4). Since such flows are restricted to lie in the exceptional divisor, we shall call them divisorial flows.

As explained after relations (2), the sign in equations (4.2) must be switched when the flow passes through a noncritical zero of . This means that the divisorial flow equations can be written in the form:

(39) |

where:

(40) |

and:

(41) |

where is now constant along each given flow
^{14}^{14}14For the examples of Section 5,
we shall take all divisorial flows to have ..

Figure 4: Picture of divisorial flows for the case .

### 4.3 General properties of divisorial flows

To understand the general properties of the flow (4.2), we must analyze the quantities and . Notice that the first function can be written in the form [24]:

(42) |

Using this observation, it is easy to see that has exactly one zero, namely (the point where meets the axis , see figure 5). Moreover, is strictly greater than zero for (the boundary of the domain of interest ) and smaller than zero for (the boundary of the complementary domain ). (In fact, coincides with the limit ). In particular, the point is a conformal infinity for the one-dimensional metric (38), as expected from the fact that the latter is the restriction of (4) to the real axis. Since we are interested in the domain , we shall restrict to in what follows.

Figure 5: The function for the model .

Expression (37) shows that has a zero at , which will belong to the region of interest only when . Notice that:

(43) |

where we used the relation . While this quantity does not vanish at , the gradient field (36) does vanish there because (and thus given in (38)) blows up at these points. Once again, this is a peculiarity of our coordinate system. It is clear from (43) that the derivative of is discontinuous at . The same is true for the gradient field . Taking , we obtain:

(44) |

(remember that is the value of corresponding to the isometry which fixes ). By property (b) of Subsection 3.1, we have , so that for . Thus is constant on if , in agreement with the fact that the isometry defined by this value of fixes the locus . Since , we find that is nonsingular (and non-negative) along this interval. It will be monotonous on the entire interval unless