Separability in Riemannian Manifolds

Separability in Riemannian Manifolds\star\star$\star$This paper is a
contribution to the Special Issue on Analytical Mechanics and Differential
Geometry in honour of Sergio Benenti. The full collection is available at
http://www.emis.de/journals/SIGMA/Benenti.html

Sergio BENENTI

S. Benenti

Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123
Torino, Italy<EMAIL_ADDRESS>http://www.sergiobenenti.it/

Received December 27, 2015, in final form January 19, 2016; Published online
February 01, 2016

An outline of the basic Riemannian structures underlying the separation of
variables in the Hamilton–Jacobi equation of natural Hamiltonian systems.

Riemannian geometry; Hamilton–Jacobi equation; separation of variables 70H20;
35F21; 58B21

This paper was submitted to the Royal Society and accepted for publication in
a special volume dedicated to the State of the Art of the Separation of
Variables $($2004$)$. However, this volume was never published due to the
sudden death of the Editor, Vadim Kutznetsov, to whom I dedicate this work.

##  1 Introduction

In the past years the study of separable systems (whose Hamilton–Jacobi
equations can be integrated by separation of variables) has shown a remarkable
development, also in relation with other kinds of integrable systems (bi-
Hamiltonian systems, Lax systems). I think that an outline of the Riemannian
background of this theory may be useful for specialists as well as beginners.
With the exception of the last section, we will confine our discussion to the
orthogonal separable systems (called Stäckel systems) and to a special class
of Stäckel systems, referred to as L-systems. Some of the theorems presented
here are new. For some of the recalled theorems a new shorter proof is
provided. Let $Q$ be an $n$-dimensional Riemannian manifold with generic local
coordinates $\underline{q}=(q^{i})$ and (contravariant) metric tensor
$\boldsymbol{G}=(g^{ij})$, which we assume to be positive-definite, and let
$T^{*}Q$ be the cotangent bundle of $Q$, with canonical coordinates
$(\underline{q},\underline{p})=(q^{i},p_{i})$. We will deal with the additive
separation of the Hamilton–Jacobi equations

$\displaystyle G(\underline{q},\underline{p})=E,\qquad
H(\underline{q},\underline{p})=E,\qquad p_{i}=\partial_{i}W,$

where $G=\tfrac{1}{2}g^{ij}p_{i}p_{j}$ is the geodesic Hamiltonian on
$T^{*}Q$, and $H=G+V=\tfrac{1}{2}g^{ij}p_{i}p_{j}+V$ is a natural Hamiltonian,
$V(\underline{q})$ being the potential energy, a smooth function on $Q$
canonically lifted to a function on $T^{*}Q$. A coordinate system
$\underline{q}$ is called separable if the geodesic Hamilton–Jacobi equation
$G=E$ admits a complete solution of the form

$\displaystyle
W(\underline{q},\underline{c})=\sum_{i=1}^{n}W_{i}\big{(}q^{i},\underline{c}\big{)},\qquad\underline{c}=(c^{i}),$
(1.1a) $\displaystyle\det\left[\dfrac{\partial^{2}W}{\partial q^{i}\partial
c^{j}}\right]\neq 0.$ (1.1b)

Such a solution is called separated solution. Also note, that in these
definitions the presence of a set of $n$ constants $\underline{c}$ satisfying
the completeness condition (1.1b) is fundamental. Note that here we consider
only natural canonical coordinates, where $\underline{q}$ are coordinates on
the configuration manifold. Levi-Civita [37] proved that a Hamilton–Jacobi
equation $H(\underline{q},\underline{p})=E$ admits a separated solution (1.1)
if and only if the differential equations111Notation:
$\partial_{i}=\partial/\partial q^{i}$, $\partial^{i}=\partial/\partial
p_{i}$.

$\displaystyle
L_{ij}(H)\doteq\partial_{i}H\partial_{j}H\partial^{i}\partial^{j}H+\partial^{i}H\partial^{j}\partial_{i}\partial_{j}H-\partial_{i}H\partial^{j}H\partial^{i}\partial_{j}H-\partial^{i}H\partial_{j}H\partial_{i}\partial^{j}H=0$

are identically satisfied. These are known as the separability conditions or
separability equations of Levi-Civita. They provide not only a simple method
for testing whether a coordinate system is separable or not, but also the
basis for the geometrical (i.e., intrinsic) characterisation of the
separation. A first (and well known) example of application is the following:
the Levi-Civita equations for a natural Hamiltonian, $L_{ij}(G+V)=0$, are
polynomial equations of fourth degree in the momenta $\underline{p}$, which
must be identically satisfied for all admissible values of these variables. It
is easy to note that the fourth-degree homogeneous part of these equations is
$L_{ij}(G)=0$. This means that: (i) the separation of the geodesic equation is
a necessary condition for the separation of equation $G+V=E$; (ii) the study
of the geodesic separation plays a prominent role, (iii) the above-given
definition of separable coordinates makes sense.

A special but fundamental case in this theory is the orthogonal separation,
where the coordinates are assumed to be orthogonal, $g^{ij}=0$ for $i\neq j$.
In this case, examined firstly by Stäckel [52, 53], later on by Levi-Civita
[37], Eisenhart [26, 27], and more recently by many authors, the Levi-Civita
equations $L_{ij}(G)=0$ are equivalent to equations

$\displaystyle S_{ij}\big{(}g^{kk}\big{)}=0,\qquad i\neq j,$ (1.2)

where $S_{ij}(\cdot)$ denote the Stäckel operators associated with an
orthogonal metric $(g^{ii})$. For any smooth function $V$ on $Q$, it is
defined by

$\displaystyle S_{ij}(V)\doteq\partial_{i}\partial_{j}V-\partial_{i}\ln
g^{jj}\partial_{j}V-\partial_{j}\ln g^{ii}\partial_{i}V,\qquad i\neq j.$

The Levi-Civita equations $L_{ij}(G+V)=0$ are equivalent to

$\displaystyle S_{ij}\big{(}g^{kk}\big{)}=0,\qquad S_{ij}(V)=0.$ (1.3)

##  2 Killing tensors

As shown by Eisenhart [26, 27] (for the orthogonal case) and by Kalnins and
Miller [32, 33], the geodesic separation is related to the existence of
Killing vectors and Killing tensors of order two. In this section we recall
the basic properties of these objects. The contravariant symmetric tensors
$\boldsymbol{K}=(K^{i\ldots j})$ on $Q$ are in one-to-one correspondence with
homogenous polynomials on $T^{*}Q$,

$\displaystyle{\boldsymbol{K}}=\big{(}K^{i\ldots j}\big{)}\
\longleftrightarrow\ P_{{\boldsymbol{K}}}=P(\boldsymbol{K})=K^{i\ldots
j}p_{i}\cdots p_{j}.$

For a tensor of order zero, i.e., a function $f$ on $Q$, we define
$P_{f}\doteq f$, where $f$ is canonically lifted to $T^{*}Q$ (by constant
values on the fibers). The space of these polynomial functions is closed with
respect to the canonical Poisson bracket

$\displaystyle\\{A,B\\}\doteq\partial^{i}A\partial_{i}B-\partial^{i}B\partial_{i}A.$

Hence, on the space of the symmetric contravariant tensors we define a Lie-
algebra structure $[\cdot,\cdot]$ by setting

$\displaystyle
P([{\boldsymbol{K}}_{1},{\boldsymbol{K}}_{2}])=\\{P({\boldsymbol{K}}_{1}),P({\boldsymbol{K}}_{2})\\},$

and the symmetric product $\odot$ by setting

$\displaystyle
P({\boldsymbol{K}}_{1}\odot{\boldsymbol{K}}_{2})=P({\boldsymbol{K}}_{1})\cdot
P({\boldsymbol{K}}_{2}).$

Note that all the above-given definitions do not depend on a metric tensor. If
a metric tensor $\boldsymbol{G}$ is present, then we say that $\boldsymbol{K}$
is a Killing tensor (KT) if $P(\boldsymbol{K})$ is in involution with
$P(\boldsymbol{G})=2G$,

$\displaystyle\\{P(\boldsymbol{K}),P(\boldsymbol{G})\\}=0\
\Longleftrightarrow\ [\boldsymbol{K},\boldsymbol{G}]=0.$ (2.1)

This means that $P(\boldsymbol{K})$ is a first integral of the geodesic flow.
In the special case of a function $f$, this definition is equivalent to
$\nabla f=0$ (by $\nabla f$ we denote the gradient of a function $f$). A
vector field $\boldsymbol{X}$ is a Killing vector,
$[\boldsymbol{X},\boldsymbol{G}]=0$, if and only if its flow preserves the
metric.

Let us consider the case of a symmetric 2-tensor $\boldsymbol{K}$. Since a
metric tensor is present, the boldface object $\boldsymbol{K}$ can be
represented in components as a tensor of type $(2,0)$, $(1,1)$ and $(0,2)$,
respectively, $\boldsymbol{K}=(K^{ij})=(K^{i}_{j})=(K_{ij})$.

As a symmetric tensor of type $(1,1)$, $\boldsymbol{K}$ defines an
endomorphism on the space $\cal X(Q)$ of the (smooth) vector fields on $Q$ and
an endomorphism on the space $\Phi^{1}(Q)$ of the (smooth) 1-forms on $Q$. We
will denote by $\boldsymbol{K}\boldsymbol{X}$ the vector field image of
$\boldsymbol{X}\in\cal X(Q)$ by $\boldsymbol{K}$, and by $\boldsymbol{K}\phi$
the 1-form image of $\phi\in\Phi^{1}(Q)$ by $\boldsymbol{K}$. This means that
$\boldsymbol{K}\boldsymbol{X}=K^{i}_{j}X^{j}\partial_{i}$,
$\boldsymbol{K}\phi=K^{i}_{j}\phi_{i}\,dq^{j}$. Note that the metric tensor
$\boldsymbol{G}$ coincide with the identity operator $\boldsymbol{I}$, whose
$(1,1)$ components are given by the Kronecker symbol $\delta^{i}_{j}$. Then a
2-tensor $\boldsymbol{K}$ gives rise to eigenvalues, eigenvectors or
eigenforms, according to equations
$\boldsymbol{K}\boldsymbol{X}=\rho\boldsymbol{X}$,
$\boldsymbol{K}\phi=\rho\phi$. We recall that, in a positive-definite metric,
(i) all symmetric tensors have real eigenvalues; (ii) the algebraic
multiplicity of an eigenvalue $\rho$ (i.e., its order as a root of the
characteristic equation $\det(\boldsymbol{K}-\rho\boldsymbol{G})=0$) is equal
to its geometrical multiplicity (i.e., the dimension of the space of the
corresponding eigenvectors, or eigenforms); the eigenspaces corresponding to
distinct eigenvalues are orthogonal. We will denote by
$\boldsymbol{K}_{1}\boldsymbol{K}_{2}$ the product of the two endomorphisms
$\boldsymbol{K}_{1}$ and $\boldsymbol{K}_{2}$; in components
$(\boldsymbol{K}_{1}\boldsymbol{K}_{2})^{ij}=K_{1}^{ih}K_{2h}^{\;j}$. The
algebraic commutator of the two tensors will be denoted by

$\displaystyle[\\![\boldsymbol{K}_{1},\boldsymbol{K}_{2}]\\!]\doteq\boldsymbol{K}_{1}\boldsymbol{K}_{2}-\boldsymbol{K}_{2}\boldsymbol{K}_{1}.$

If a symmetric 2-tensor $\boldsymbol{K}$ can be diagonalised in orthogonal
coordinates, $K^{ij}=0$ for $i\neq j$, then $K^{ii}=\rho^{i}g^{ii}$, where
$(\rho^{i})$ are the eigenvalues of $\boldsymbol{K}$. By writing the Killing
equation (2.1) in these coordinates, we see that $\boldsymbol{K}$ is a KT if
and only if equations

$\displaystyle\partial_{i}\rho^{j}=\big{(}\rho^{i}-\rho^{j}\big{)}\partial_{i}\ln
g^{jj}.$ (2.2)

are satisfied by the eigenvalues. These equations have been called
Killing–Eisenhart equations in [13] since they have been extensively used by
Eisenhart [27]. However, they appear in an earlier paper by Levi-Civita [36,
p. 285].

##  3 Killing–Stäckel spaces

Equations (2.2) can be interpreted as a linear system of $n$ first-order
partial differential equations in normal form, in the $n$ unknown functions
$\rho^{i}(\underline{q})$. It is a remarkable fact that the integrability
conditions assume of the form

$\displaystyle\big{(}\rho^{i}-\rho^{j}\big{)}S_{ij}\big{(}g^{kk}\big{)}=0.$

Then their link with the orthogonal separation is at once clear. A second, and
even more remarkable property, is that the unknown functions $\rho^{i}$ appear
in the integrability conditions through their differences $\rho^{i}-\rho^{j}$.
This means that if the system (2.2) admits a solution such that
$\rho^{i}\neq\rho^{j}$, then it is completely integrable. Note that for a
linear system the converse is always true. Going back to the orthogonal
separability conditions (1.2) we can immediately conclude that: (I) a system
of orthogonal coordinates is separable if and only if there exists a KT which
is diagonalised in these coordinates and which has pointwise simple
eigenvalues. Furthermore, since the system (2.2) is linear, if it is
completely integrable then it admits a $n$-dimensional space of solutions (and
its converse). As a consequence: (II) a Killing tensor $\boldsymbol{K}$ which
has simple eigenvalues and is diagonalised in orthogonal coordinates generates
a $n$-dimensional space $\cal K$ of Killing tensors which are all diagonalised
in the same coordinates. Such a space will be called Killing–Stäckel space
(KS-space). In the space of direct sums of Killing tensors,
${\boldsymbol{K}}=c\oplus\boldsymbol{K}_{1}\oplus\boldsymbol{K}_{2}\oplus\cdots\oplus\boldsymbol{K}_{n}\oplus\cdots$
endowed with the Lie bracket $[\cdot,\cdot]$ defined above, a KS-space, which
is made of elements $0\oplus 0\oplus\boldsymbol{K}_{2}\oplus 0\oplus\cdots$,
is an involutive subalgebra. For this reason it has also been called
Killing–Stäckel algebra in [13, 14].

Three remarks are in order: (i) the metric tensor belongs to any KS-space –
indeed, $\rho^{i}=1$ is a trivial solution of the system (2.2); (ii) if two
KS-spaces have an element with simple eigenvalues in common, then they
coincide; (iii) all elements of a KS-space are in involution (if equations
(2.2) are satisfied for two tensors $\boldsymbol{K}_{1}$ and
$\boldsymbol{K}_{2}$, then
$\\{P(\boldsymbol{K}_{1}),P(\boldsymbol{K}_{2})\\}=0$).

All the above properties have a local character and are related to a
coordinate system. We remark, however, that they are more precisely related to
an equivalence class of orthogonal systems, being equivalent to two systems of
coordinates $\underline{q}$ and $\underline{q}^{\prime}$ simply related by a
separated transformation or a rescaling: $q^{i}=q^{i}(q^{i^{\prime}})$.

We look for a coordinate-free description of all this matter. To this end we
recall some basic concepts.

A frame on a differentiable manifold $Q$ (not necessarily Riemannian) is a set
of vector fields $(\boldsymbol{X}_{i})$ which form a basis of the tangent
space $T_{q}Q$ at each point $q$ of their domain of definition. In general,
frames exist only locally. Global frames are defined if and only if the
manifold is parallelisable, i.e., when $TQ\simeq Q\times\mathbb{R}^{n}$. Two
frames $(\boldsymbol{X}_{i})$ and $(\boldsymbol{X}^{\prime}_{i})$ are said to
be equivalent if there are nowhere vanishing functions $f_{i}$ such that
$\boldsymbol{X}_{i}=f_{i}\boldsymbol{X}^{\prime}_{i}$. A frame is called
holonomic or integrable if it is equivalent to a natural frame
$(\partial_{i})$ associated with coordinates $(q^{i})$. A basic property is
(cf. [49] and [13])

######  Theorem 3.1.

The three following conditions are equivalent: $(i)$ the frame
$(\boldsymbol{X}_{i})$ is holonomic, $(ii)$ for each pair of indices $(i,j)$
the distribution spanned by the vectors $\boldsymbol{X}_{i}$ and
$\boldsymbol{X}_{j}$ is completely integrable, $(iii)$ for each index $i$ the
distribution spanned by the $n-1$ vectors $\boldsymbol{X}_{j}$ for $j\neq i$
is completely integrable.

On a Riemannian manifold a vector field $\boldsymbol{X}$ is called normal if
it is orthogonally integrable or surface forming, i.e., if it is orthogonal to
a family of hypersurfaces. In a positive-definite metric a symmetric tensor
$\boldsymbol{K}$ with simple eigenvalues and normal eigenvectors gives rise to
and equivalence class of holonomic orthogonal frames hence, to an equivalence
class of orthogonal coordinates. Then we get the following simple intrinsic
characterisation of the orthogonal geodesic separation [32, Corollary 4,
Section 3] (see also [6]):

######  Theorem 3.2.

The geodesic Hamilton–Jacobi equation is separable in orthogonal coordinates
if and only if there exists a Killing $2$-tensor with simple eigenvalues and
normal eigenvectors.

A KT having these properties will be called a characteristic Killing tensor
(ChKT). The eigenvectors generates a family of $n$ orthogonal foliations of
manifolds of codimension 1, which we call Stäckel web. Any coordinate system
$(q^{i})$ such that the web is locally described by equations $q^{i}={\rm
const}$ (this is equivalent to say that $dq^{i}$ are eigenforms of the ChKT),
is separable.

In accordance with the remarks above, a ChKT generates a KS-space. A suitable
coordinate-independent definition of this concept is the following: a Stäckel
space on a Riemannian manifold $Q_{n}$ is a $n$-dimensional linear space
${\cal K}_{n}$ of Killing 2-tensors whose elements (a) commute as linear
operators, $[\\![\boldsymbol{K}_{1},\boldsymbol{K}_{2}]\\!]=0$, and (b) are in
involution, $[\boldsymbol{K}_{1},\boldsymbol{K}_{2}]=0$. Indeed, in the
algebraic realm it can be proved that in such a space there exists an element
with pointwise distinct eigenvalues (in the neighborhood of any given point of
the domain of definition of ${\cal K}_{n}$); as a consequence, the commutation
relation (a), applied to such a tensor $\boldsymbol{K}_{1}$, shows that all
elements have common eigenvectors. Furthermore, from (b) it follows that

######  Theorem 3.3.

If $n$ independent KT’s in involution have the same eigenvectors, then these
eigenvectors are normal.

This remarkable property was firstly discovered by Kalnins and Miller [32].
However, it is also a remarkable fact that in this last theorem the assumption
that the independent tensors are KTs is redundant. In fact, it can be proved
that

######  Theorem 3.4.

An orthogonal frame made of common eigenvectors of $n$ independent symmetric
$2$-tensors in involution is holonomic $($the eigenvectors are normal$)$.

For a detailed discussion and proof see [13]. As a consequence, we have a
second intrinsic characterisation of the orthogonal geodesic separation
(compare with [32, Theorem 6, Section 3] and [50]; note that in [32, Theorem
6(4)] turns out to be redundant):

######  Theorem 3.5.

The geodesic Hamilton–Jacobi equation is separable in orthogonal coordinates
if and only if the Riemannian manifold admits a KS-space, i.e., a
$n$-dimensional linear space ${\cal K}$ of Killing tensors commuting as linear
operators and in involution.

In the applications, one of these last two theorems can be used according to
the convenience. By applying Theorem 3.2 we have the advantage of dealing with
a single KT, but difficulties may arise in testing if it has simple
eigenvalues and normal eigenvectors. Nevertheless, for solving this problem we
can use the following two effective criteria:

######  Theorem 3.6.

A $(1,1)$ tensor $\boldsymbol{K}$ has distinct eigenvalues if and only if

$\displaystyle D\doteq\begin{vmatrix}n&S_{1}&\ldots&S_{n-1}\cr
S_{1}&S_{2}&\ldots&S_{n}\cr\vdots&\vdots&\cdots&\vdots\cr
S_{n-1}&S_{n}&\ldots&S_{2n-2}\end{vmatrix}\neq 0,\qquad
S_{p}\doteq\operatorname{tr}\big{(}\boldsymbol{K}^{p}\big{)}.$

Here $\boldsymbol{K}^{p}$ is the power $p=0,1,2,\ldots$ of the linear mapping
$\boldsymbol{K}$. This theorem is a consequence of a classical theorem of
Sylvester about the discriminant $D$ of an algebraic equation, here applied to
the characteristic equation of $\boldsymbol{K}$.

######  Theorem 3.7.

A symmetric tensor $\boldsymbol{K}$ with simple eigenvalues has normal
eigenvectors if and only if

$\displaystyle
H_{ab}^{h}K^{a}_{i}K^{b}_{j}+2H_{a[i}^{b}K^{a}_{j]}K^{h}_{b}+H^{a}_{ij}K^{h}_{b}K^{b}_{a}=0,$

where $H$ is the Nijenhuis torsion of $\boldsymbol{K}$,

$\displaystyle H_{ij}^{h}(\boldsymbol{K})\doteq
2K^{a}_{[i}\partial_{|a|}K_{j]}^{h}-2K^{h}_{a}\partial_{[i}K^{a}_{j]}.$

This is a special case of a more general theorem due to Haantjes [29] (see
also [49, p. 248]). As will be seen in Section 7, it is interesting the case
of a torsionless tensor: $\boldsymbol{H}(\boldsymbol{K})=0$. We will apply the
following

######  Theorem 3.8 ([45]).

A symmetric tensor $\boldsymbol{K}$ with simple eigenvalues $\rho^{i}$ is
torsionless if and only if it has normal eigenvectors $\boldsymbol{X}_{i}$
such that

$\displaystyle\boldsymbol{X}_{i}\rho^{j}=0,\qquad i\neq j.$

This means that each eigenvalue $\rho^{j}$ is constant on the hypersurfaces
orthogonal to the corresponding eigenvector $\boldsymbol{X}_{j}$. It is
worthwhile to observe that

######  Theorem 3.9.

A torsionless KT with simple eigenvalues is necessarily a constant KT on a
flat Riemannian manifold.

###### Proof.

If $\boldsymbol{H}(\boldsymbol{K})=0$, then equations (2.2) imply

$\displaystyle
0=\partial_{i}\rho^{j}=\big{(}\rho^{i}-\rho^{j}\big{)}\partial_{i}\ln
g^{jj},\qquad i\neq j,\qquad\partial_{i}\rho^{i}=0.$

This shows that $\rho^{i}={\rm const}$ and that $\partial_{i}g^{jj}=0$ for
$i\neq j$. This last condition means that $g^{jj}$ is a function of $q^{j}$
only. In this case, up to a change of scale of the coordinates, we can
consider $g^{jj}={\rm const}$. ∎

This is a case considered in [21]. Going back to Theorem 3.5, it is
interesting to make a comparison with the intrinsic characterisations of the
geodesic orthogonal separation due to Eisenhart [26, 27] and Woodhouse [58].
In the Eisenhart theorem [26, p. 289] the necessary and sufficient conditions
for the orthogonal separation are: (i) the existence of $n-1$ independent
Killing tensors $\boldsymbol{K}_{1},\ldots,\boldsymbol{K}_{n-1}$ with normal
common eigenvectors and such that (ii) for each of these tensors the
eigenvalues are simple and (iii) for any pair $(i,\alpha)$ of fixed indices
($\alpha=2,\ldots,n$, $i=1,\ldots,n$) the square matrices
$\|\rho_{i}^{\alpha}-\rho_{j}^{\alpha}\|$ (with $j\neq i$) are regular. In the
Eisenhart notation, $\rho_{i}^{\alpha}$ are the eigenvalues of
$\boldsymbol{K}_{\alpha}$. Condition (i) should be replaced by the existence
of $n-1$ Killing tensors such that
$\boldsymbol{G},\boldsymbol{K}_{1},\ldots,\boldsymbol{K}_{n-1}$ are
independent. Then Theorem 3.5 shows that conditions (ii) and (iii) are
redundant. In [58, Theorem 4.2] the $n-1$ KT’s are assumed to be in involution
and with common closed eigenforms. Theorem 3.3 shows that the requirement
closed is redundant, since it is equivalent to the normality of the
eigenvectors.

##  4 The orthogonal separation of a natural Hamiltonian

With each symmetric 2-tensor $\boldsymbol{K}$ and a function $U$ on $Q$ we
associate the function $F=\tfrac{1}{2}P_{\boldsymbol{K}}+U$ on $T^{*}Q$. We
observe that $F$ is a first integral of $H=G+V$, $\\{H,F\\}=0$, if and only if

$\displaystyle\\{G,P_{\boldsymbol{K}}\\}=0,\qquad dU=\boldsymbol{K}\,dV.$

The first equation means that $\boldsymbol{K}$ is a Killing tensor. If it is a
ChKT, then, in any orthogonal coordinate system determined by its
eigenvectors, the second equation is equivalent to
$\partial_{i}U=\rho^{i}\partial_{i}V$. Due to the fundamental equations (2.2),
the integrability conditions of these equations assume the form
$\partial_{j}\partial_{i}U-\partial_{i}\partial_{j}U=(\rho^{i}-\rho^{j})S_{ij}(V)=0$.
This proves

######  Theorem 4.1.

If $\boldsymbol{K}$ is a symmetric $2$-tensor with simple eigenvalues and
normal eigenvectors, then $F$ is a first integral of $G+V$ if and only if
$\boldsymbol{K}$ is a Killing tensor and $S_{ij}(V)=0$ in any orthogonal
system of coordinates generated by the eigenvectors.

Since the existence of coordinates is a local matter, condition
$dU=\boldsymbol{K}\,dV$ can be replaced by $d(\boldsymbol{K}\,dV)=0$. Thus, by
recalling Theorem 3.2 and the remarks at the end of Section 1, we find that

######  Theorem 4.2 ([6]).

The Hamilton–Jacobi equation $G+V=E$ is separable in orthogonal coordinates if
and only if there exists a Killing $2$-tensor $\boldsymbol{K}$ with simple
eigenvalues and normal eigenvectors such that

$\displaystyle d(\boldsymbol{K}\,dV)=0.$ (4.1)

This equation has been called characteristic equation of a separable potential
$V$. We observe that for $n=2$ any vector field is normal. Since it can be
proved that on a two-dimensional manifold the separation always occurs in
orthogonal coordinates [37], we get

######  Theorem 4.3.

On a two-dimensional Riemannian manifold the Hamilton–Jacobi equation $G+V=E$
is separable if and only if there exists a $($non-trivial$)$ quadratic first
integral.

This is the extension to a two-dimensional manifold of the so-called
Bertrand–Darboux–Whittaker theorem for the Euclidean plane $\mathbb{E}_{2}$
[1, 57].

When written in Cartesian coordinates on a Euclidean $n$-space, equation (4.1)
gives rise to the so-called Bertrand–Darboux (BD) equations. If we know the
form of all characteristic tensors of a manifold, then equation (4.1), written
in any coordinate system (even not separable), provides an effective criterion
for the separability of the Hamilton–Jacobi equation. This criterion have been
applied for instance in the study of the super-separability of the inverse-
square three-dimensional Calogero system [11]. In [11] you can find the
intrinsic (i.e., “boldface”) expressions of all the characteristic Killing
2-tensors in the Euclidean three-space $\mathbb{E}_{3}$, so that this
criterion is ready to be used for any potential $V$. For a general
$\mathbb{E}_{n}$, the basic ChKT’s and the corresponding BD equations have
been determined in Marshall and Wojciechowski, and in [4, 6]. This analysis
has been completed by Waksjö [55]. In his thesis he presents an effective
general criterion for the separability of a potential $V$ in the Euclidean
$n$-space (see also [56]). Other separability criteria can be based on the
analysis of the fundamental invariants of spaces of Killing tensors under the
action of isometry groups and the method of moving frames (see [24, 42], also
for related references).

If in $\mathbb{E}_{n}$ we take

$\displaystyle\boldsymbol{K}=\text{tr}(\boldsymbol{L})\boldsymbol{G}-\boldsymbol{L},\qquad\boldsymbol{L}\doteq\boldsymbol{A}+m\boldsymbol{r}\otimes\boldsymbol{r}+\boldsymbol{w}\odot\boldsymbol{r},$
(4.2)

where $\boldsymbol{A}$ is symmetric and constant, $m\in\mathbb{R}$,
$\boldsymbol{w}$ is a constant vector, and $\boldsymbol{r}$ is the vector
representing the generic point, then in Cartesian coordinates $(x^{i})$
equation (4.1) yields the Bertrand–Darboux equations (3.25) in [55]. The
correspondence of notation is the following: $\boldsymbol{A}=(\gamma_{ij})$,
$m=\alpha$, $\boldsymbol{w}=(2\beta_{i})$).

For $\boldsymbol{w}=0$ we get the BD equations for the separation in elliptic
coordinates centered at the origin. For $m=0$ we get the BD equations for the
separation in parabolic coordinates centered at the point $P$, where
$\boldsymbol{L}_{P}(\boldsymbol{w})$ (the existence of such a point is proved
in [4]). For $\boldsymbol{w}=0$ and $m=0$ we have the separation in Cartesian
coordinates. In the remaining case $\boldsymbol{w}\neq 0$ and $m\neq 0$, we
have the separation in elliptic coordinates centered at the point
$\boldsymbol{c}=-\tfrac{1}{2m}\boldsymbol{w}$ (see below).

It must be emphasised that in the characteristic equation (4.1) for a
separable potential $V$ the eigenvalues of $\boldsymbol{K}$ must be simple
(outside a singular set). For the present case we have

######  Theorem 4.4.

The tensor $\boldsymbol{K}$ defined in (4.2) has simple eigenvalues if and
only if for $m=0$ the eigenvalues of $\boldsymbol{A}$ are simple and for
$m\neq 0$ the eigenvalues of
$\boldsymbol{A}-\tfrac{1}{4m}\boldsymbol{w}\otimes\boldsymbol{w}$ are simple.

This is (a slightly modified version of) a theorem of Waksjö [55, p. 45]. We
give here an alternative proof.

###### Proof.

The tensor $\boldsymbol{K}$ has simple eigenvalues if and only if
$\boldsymbol{L}$ has simple eigenvalues. For $m=0$ we have
$\boldsymbol{L}=\boldsymbol{A}+\boldsymbol{w}\odot\boldsymbol{r}$; we are in
the case of the parabolic web [4] and $\boldsymbol{L}$ has simple eigenvalues
if and only if $\boldsymbol{A}$ has simple (constant) eigenvalues. For $m\neq
0$, let us change the origin by considering the transformation
$\boldsymbol{r}=\boldsymbol{c}+\boldsymbol{r}^{\prime}$, where
$\boldsymbol{c}$ is a constant vector. We obtain

$\displaystyle\boldsymbol{L}=\boldsymbol{A}+m(\boldsymbol{r}^{\prime}\otimes\boldsymbol{r}^{\prime}+2\boldsymbol{r}^{\prime}\odot\boldsymbol{c}+\boldsymbol{c}\otimes\boldsymbol{c})+\boldsymbol{w}\odot\boldsymbol{c}+\boldsymbol{w}\odot\boldsymbol{r}^{\prime}.$

If we take $\boldsymbol{c}=-\tfrac{1}{2m}\boldsymbol{w}$, then

$\displaystyle\boldsymbol{L}=\boldsymbol{A}-\tfrac{1}{4m}\boldsymbol{w}\otimes\boldsymbol{w}+m\boldsymbol{r}^{\prime}\otimes\boldsymbol{r}^{\prime}=\boldsymbol{A}^{\prime}+m\boldsymbol{r}^{\prime}\otimes\boldsymbol{r}^{\prime}.$

We known that for a symmetric tensor of the kind
$\boldsymbol{A}^{\prime}+\boldsymbol{r}^{\prime}\otimes\boldsymbol{r}^{\prime}$
the eigenvalues $(u^{i})$ are the roots of equation

$\displaystyle\sum_{i}\dfrac{x_{i}^{2}}{u-a^{i}}=\frac{1}{m},$

where $(x_{i})$ are Cartesian coordinates and $(a^{i})$ are the constant
eigenvalues of $\boldsymbol{A}^{\prime}$. This equation is equivalent to

$\displaystyle m\sum_{i}x_{i}^{2}\prod_{k\neq
i}\big{(}u-a^{k}\big{)}-\prod_{j}\big{(}u-a^{j}\big{)}=0.$ (4.3)

If $(a^{i})$ are simple, then also $(u^{i})$ are simple, since
$a^{1}<u^{1}<a^{2}<u^{2}<\ldots<u^{n-1}<a^{n}<u^{n}$. If $(a^{i})$ are not all
simple, for instance $a^{1}=a^{2}$, then equation (4.3) has a double root
$u=a^{1}$. ∎

The tensors $\boldsymbol{K}$ and $\boldsymbol{L}$ have a mechanical meaning:
they were introduced in [4] as the inertia tensor and planar inertia tensor of
a set of massive points (including, this is important, negative masses) in
$\mathbb{E}_{n}$. The parameter $m$ is just the total mass (it may be $0$).
Indeed, it is a remarkable fact that an inertia tensor is a KT. This
interpretation is of help in the problem of finding the intrinsic expressions
of all the ChKT’s of $\mathbb{E}_{n}$ (see also [41]).

######  Notation 4.5.

A matrix of the kind $\boldsymbol{A}+\boldsymbol{r}\otimes\boldsymbol{r}$ has
been used by Moser [44] for constructing a Lax pair for the geodesic flow of
an asymmetric ellipsoid. For this reason it was denoted by $\boldsymbol{L}$ in
[4] (indeed, in analogy with the Lax method, starting from $\boldsymbol{L}$ we
can construct a complete system of first integrals in involution through a
pure algebraic process). There were other two reasons which suggested this
notation: (i) $\boldsymbol{L}$ is a letter adjacent to $\boldsymbol{K}$, and
this is appropriate because a tensor $\boldsymbol{L}$ generates a tensor
$\boldsymbol{K}$ according to (4.2); (ii) $\boldsymbol{L}$ stands for Levi-
Civita, and indeed the orthogonal metric associated with $\boldsymbol{L}$ was
firstly introduced by Levi-Civita [36] (see the end of Section 7).

##  5 First integrals associated with the orthogonal separation

Going back to the characteristic equation (4.1) we recall that at the
beginning of Section 3 we observed that a characteristic Killing tensor, like
that appearing in equation (4.1), generates a KS-space $\cal K$. It is a
remarkable fact that

######  Theorem 5.1.

If
$(\boldsymbol{K}_{a})=(\boldsymbol{K}_{0},\boldsymbol{K}_{2},\ldots,\boldsymbol{K}_{n-1})$
is a basis of $\cal K$ $(\boldsymbol{G}$ and $\boldsymbol{K}$ may belong to
this basis$)$ then locally there exist functions $V_{a}$ such that

$\displaystyle H_{a}\doteq\tfrac{1}{2}P_{\boldsymbol{K}_{a}}+V_{a}$ (5.1)

are independent first integrals in involution.

A proof can be found in [6]. Indeed, it can be shown that equation (4.1)
implies $d(\boldsymbol{K}_{a}\,dV)=0$ for each index $a$. This implies that a
ChKT $\boldsymbol{K}$ and a separable potential $V$ generates a
$n$-dimensional space ${\cal H}(\boldsymbol{K},V)$ of first integrals in
involution,

$\displaystyle
H_{\boldsymbol{K}}=\tfrac{1}{2}P_{\boldsymbol{K}}+V_{\boldsymbol{K}},\qquad\boldsymbol{K}\in\cal
K,$

and that the associated potentials $V_{\boldsymbol{K}}$ can be determined by
integrating the closed 1-forms $\boldsymbol{K}_{a}\,dV$. We observe that there
are separable systems (an example is the three-body Calogero system, see [11])
in which this integration can be avoided and replaced by an algebraic process.

We remark that, if we know a basis of a KS-space, then for testing if a
potential $V$ is separable it is sufficient to verify that equation (4.1) is
satisfied for an element of this space with simple eigenvalues. When the
answer is affirmative, then a complete set of integrals in involution can be
determined by integrating the closed 1-forms $\boldsymbol{K}_{a}\,dV$.

It is well known that the orthogonal (as well as the non-orthogonal
separation) is related to Stäckel matrices. A Stäckel matrix in the $n$
variables $(q^{i})$ is a regular $n\times n$ matrix
$\boldsymbol{S}=\big{[}\varphi_{i}^{(a)}\big{]}$ of functions
$\varphi_{i}^{(a)}$ depending on the variable $q^{i}$ corresponding to the
lower index only. We denote by $\big{[}\varphi_{(a)}^{i}\big{]}$ the inverse
matrix. The original Stäckel theorem asserts that an orthogonal coordinate
system $(q^{i})$ is separable if and only if there exists a Stäckel matrix
such that $g^{ii}=\varphi_{(0)}^{i}$. In this case, (i) the diagonalised
tensors $K_{a}^{ii}=\varphi_{(a)}^{i}$ are the basis of a Stäckel space, (ii)
all separable potentials $V$ have the form
$V=\phi_{i}(q^{i})\varphi_{(0)}^{i}$, where $\phi_{i}$ is a function of the
corresponding coordinate $q^{i}$ only; (iii) a basis $H_{a}$ of the space of
first integrals in involution is given by (5.1) with
$V_{a}=\phi_{i}\varphi_{(a)}^{i}$.222Stäckel systems have been the object of
several researches in recent years. A generalisation of Stäckel systems has
been proposed in [19]. Relations between finite- (Stäckel systems) and
infinite-dimensional (Harry–Dym hierarchies) integrable systems are
investigated in [40]. The separability of a Hamiltonian in different Stäckel
systems is a standard procedure to determine superintegrable systems (see [43]
and references therein). More links between Stäckel systems and
superintegrability are shown in [54]. An analysis of the Killing–Stäckel
spaces is the basis for a classification of the orthogonal separable
coordinates in [46]. (Bibliographical footnotes added by Giovanni Rastelli.)

##  6 Conformal Killing tensors

As remarked above, the existence of a KT $\boldsymbol{K}$ with simple
eigenvalues and normal eigenvectors is a necessary and sufficient condition
for the existence of a KS-space $\cal K$, i.e., of a $n$-dimensional linear
space of KT’s with common normal eigenvectors (and, consequently, in
involution). The following question arises: is it possible to construct a
basis of the space $\cal K$ by a coordinate independent algebraic procedure,
starting from $\boldsymbol{K}$? Note that this problem can be solved (in
principle) by integrating the linear differential system (2.2), if we know a
separable coordinate system.

As illustrated in [4] the answer is affirmative at least for special kinds of
Stäckel systems. In the next sections we will revisit this matter, by
proposing new definitions and theorems. To this end, we need to recall some
basic definitions and properties concerning conformal Killing tensors.

A conformal Killing tensor (CKT) on a Riemannian manifold $Q_{n}$ is a
symmetric tensor $\boldsymbol{L}$ of order $l$ satisfying equation
$\\{P_{\boldsymbol{L}},P_{\boldsymbol{G}}\\}=P_{\boldsymbol{X}}P_{\boldsymbol{G}}$,
where $\boldsymbol{X}$ is a suitable symmetric tensor of order $l-1$. Since we
are interested in CKT’s of order two, we write this equation in the form

$\displaystyle\\{P_{\boldsymbol{L}},P_{\boldsymbol{G}}\\}=-2P_{\boldsymbol{C}}P_{\boldsymbol{G}},$
(6.1)

where $\boldsymbol{C}$ is a vector field which we call associated with
$\boldsymbol{L}$ (also denoted by $\boldsymbol{C}(\boldsymbol{L})$).

A CKT is said to be of gradient-type (GCKT) if $\boldsymbol{C}=\nabla f$. An
example of GCKT is $f\boldsymbol{G}$. In this case we have
$\boldsymbol{C}=\nabla f$. Indeed,
$\\{P_{f\boldsymbol{G}},P_{\boldsymbol{G}}\\}=\\{fP_{\boldsymbol{G}},P_{\boldsymbol{G}}\\}=P_{\boldsymbol{G}}\\{f,P_{\boldsymbol{G}}\\}=-2g^{ij}p_{j}\partial_{i}fP_{\boldsymbol{G}}$.
A KT is obviously a GCKT with $\boldsymbol{C}=0$.

######  Theorem 6.1.

A CKT $\boldsymbol{L}$ is of gradient-type with $\boldsymbol{C}=\nabla f$ if
and only if $\boldsymbol{K}=f\boldsymbol{G}-\boldsymbol{L}$ is a Killing
tensor.

###### Proof.

$\\{P_{\boldsymbol{L}}-fP_{\boldsymbol{G}},P_{\boldsymbol{G}}\\}=\\{P_{\boldsymbol{L}},P_{\boldsymbol{G}}\\}-P_{\boldsymbol{G}}\\{f,P_{\boldsymbol{G}}\\}=-2P_{\boldsymbol{C}}P_{\boldsymbol{G}}+2P_{\boldsymbol{G}}P_{\nabla
f}$. ∎

A conformal Killing tensor of trace-type $\boldsymbol{L}$ is a CKT for which
$\boldsymbol{C}=\nabla\operatorname{tr}(\boldsymbol{L})$. In this case,
$\boldsymbol{K}=\operatorname{tr}(\boldsymbol{L})\boldsymbol{G}-\boldsymbol{L}$
is a KT.

By considering in (6.1) $P_{\boldsymbol{G}}=g^{ii}p_{i}^{2}$ and
$P_{\boldsymbol{L}}=u^{i}g^{ii}p_{i}^{2}$, we find

######  Theorem 6.2.

Assume that a symmetric $2$-tensor $\boldsymbol{L}$ is diagonalised in
orthogonal coordinates, so that $L^{ii}=u^{i}g^{ii}$, $L^{ij}=0$, $i\neq j$,
where $u^{i}$ are the eigenvalues. Then $\boldsymbol{L}$ is a CKT if and only
if the following equations are satisfied,

$\displaystyle\partial_{i}u^{k}=\big{(}u^{i}-u^{k}\big{)}\partial_{i}\ln
g^{kk}+C_{i},\qquad C_{i}=\partial_{i}u^{i}.$ (6.2)

##  7 L-tensors

Let us call L-tensor a (i) conformal Killing tensor $\boldsymbol{L}$ with (ii)
vanishing torsion and (iii) pointwise simple eigenvalues $(u^{i})$. The
reasons for introducing such an object will be explained in the next section.
In the present section we examine the basic properties of an L-tensor. Due to
the vanishing of the torsion, there is an equivalence classes of orthogonal
coordinates $(q^{i})$ in which this tensor is diagonalised and
$\partial_{i}u^{j}=0$ for $i\neq j$. We say that these coordinates are
associated with $\boldsymbol{L}$. Since $\boldsymbol{L}$ is a CKT, due to
Theorem 6.2 equations

$\displaystyle\partial_{i}u^{j}=\big{(}u^{i}-u^{j}\big{)}\partial_{i}\ln
g^{jj}+C_{i}=0,\qquad i\neq j,\qquad C_{i}=\partial_{i}u^{i}$ (7.1)

hold. For $\boldsymbol{C}=0$ we find the equations (2.2) of a KT.

######  Remark 7.1.

In the above definition no assumption is made about the independence of the
eigenvalues as functions on $Q$; some of them may be constant (a criterion for
the independence of the eigenvalues is given in Theorem 9.2 below). This
definition has to be compared with those given in [15, 20, 30], where
$\boldsymbol{L}$ is assumed to be a torsionless CKT of trace-type (and called
Benenti tensor), and in [22, 23], where $\boldsymbol{L}$ is assumed to be a
torsionless CKT with independent (i.e., coordinate-forming) eigenvalues (and
called special conformal Killing tensor, see Theorem 9.1 below). In all these
papers the essential condition that $\boldsymbol{L}$ has simple eigenvalues is
missing (or understood).

In fact it can be proved that

######  Theorem 7.2.

If an eigenvalue $u^{i}$ of a torsionless CKT of trace-type is not simple,
then it is constant.

The proof of this theorem (here omitted) requires the use of the Haantjes
theorem for a tensor with non-simple eigenvalues.

######  Theorem 7.3.

Let $\boldsymbol{L}$ be an L-tensor with associated coordinates $(q^{i})$.
Then: $(i)$ Each eigenvalue $u^{i}$ depends on the associated coordinate
$q^{i}$ only, $u^{i}=u^{i}(q^{i})$. $(ii)$ It is of trace-type,
$\boldsymbol{C}=\nabla\operatorname{tr}(\boldsymbol{L}).$ $(iii)$ In
associated coordinates the metric has the form

$\displaystyle g^{kk}=\phi_{k}\prod_{i\neq k}\dfrac{1}{|u^{i}-u^{k}|},\qquad
u^{i}=u^{i}\big{(}q^{i}\big{)},\qquad\phi_{k}=\phi_{k}\big{(}q^{k}\big{)}>0,$
(7.2)

or, after a rescaling

$\displaystyle g^{kk}=\prod_{i\neq k}\dfrac{1}{|u^{i}-u^{k}|}.$ (7.3)

In both cases,

$\displaystyle\partial_{i}\ln
g^{kk}=\dfrac{\partial_{i}u^{i}}{u^{k}-u^{i}},\qquad i\neq k.$

$($We call normal coordinates associated with $\boldsymbol{L}$ the orthogonal
coordinates for which equations (7.3) hold$)$. $(iv)$ The associated
coordinates are separable. $(v)$ $\boldsymbol{L}$ commutes with the Ricci
tensor $\boldsymbol{R}$, $[\\![\boldsymbol{L},\boldsymbol{R}]\\!]=0$, i.e.,
the Robertson condition is satisfied: in the associated coordinates,
$R_{ij}=0$ for $i\neq j$.

These properties are derived from [4, 6]. They follow from equations (7.1) and
from the fundamental properties of the elementary symmetric polynomials (see
the next section). From (7.2) it follows that the contracted Christoffel
symbols $\Gamma_{i}=g^{hj}\Gamma_{hj,i}$ take the simple form
$\Gamma_{i}=-\tfrac{1}{2}\phi^{\prime}_{k}\phi_{k}$. Thus, in normal
coordinates $\Gamma_{i}=0$. Since in separable orthogonal coordinates
$R_{ij}=\tfrac{3}{2}\partial_{i}\Gamma_{j}$, for $i\neq j$, the Robertson
condition (v) is proved. Item (ii) (which follows from (7.1):
$C_{i}=\partial_{i}u^{i}$ and $\partial_{i}u^{j}=0$ for $i\neq j$ implies
$C_{i}=\partial_{i}\sum_{j}u^{j}$) shows that in [22, Proposition 1, Section
1] the assumption that $(u^{i})$ are functionally independent eigenvalues is
redundant.

It is a remarkable fact that the metric (7.2) associated with $\boldsymbol{L}$
is that of the corresponding geodesics found by Levi-Civita [36]: there exists
a metric $\overline{\boldsymbol{G}}=(\overline{g}^{ij})$ having the same
$($unparametrized$)$ geodesics of a given metric $\boldsymbol{G}=(g^{ij})$ if
and only if there are orthogonal coordinates in which the metric
$\boldsymbol{G}$ assumes the form (7.2). It must be pointed out that this
theorem holds under the assumption that the tensor $\overline{\boldsymbol{G}}$
has simple eigenvalues with respect to $\boldsymbol{G}$. This matter has been
recently analysed by Bolsinov and Matveev [20] and by Crampin [23]. The metric
(7.2) is a special case of the orthogonal separable metric determined by
Eisenhart [27, Appendix 13] and characterised by the condition $R_{jiik}=0$
for $i,j,k\neq$.

##  8 L-systems

Let $\sigma_{a}(\underline{u})$ denote the elementary symmetric polynomial of
degree $a$ of the $n$ variables $\underline{u}=(u^{i})$. Let $\sigma_{a}^{i}$
and $\sigma_{a}^{ij}$ be the functions obtained from $\sigma_{a}$ by setting
$u^{i}=0$ and $u^{j}=0$. Let us set

$\displaystyle\sigma_{0}=\sigma_{0}^{i}=\sigma_{0}^{ij}=1,\qquad\sigma_{-1}=\sigma_{-1}^{i}=\sigma_{-1}^{ij}=0,\qquad\sigma^{ij}_{n}=\sigma_{n-1}^{ij}=0.$
(8.1)

Then the following equations are satisfied [4, Section 2]:

$\displaystyle\sigma_{a}=\sigma_{a}^{i}+u^{i}\sigma_{a-1}^{i},\qquad\sigma_{a}^{j}=\sigma_{a}^{ij}+u^{i}\sigma_{a-1}^{ij},\qquad\sigma_{a}^{i}-\sigma_{a}^{j}=\big{(}u^{j}-u^{i}\big{)}\sigma^{ij}_{a-1}.$
(8.2)
$\displaystyle\sum_{i}u^{i}\sigma_{a-1}^{i}=a\sigma_{a},\qquad\det\big{[}\sigma_{a}^{i}\big{]}=\prod_{j>i}\big{(}u^{i}-u^{j}\big{)}.$
(8.3) $\displaystyle\dfrac{\partial\sigma_{a}}{\partial
u^{i}}=\sigma^{i}_{a-1},\qquad\dfrac{\partial\sigma_{a}^{j}}{\partial
u^{i}}=\sigma_{a-1}^{ij},\qquad\big{(}u^{j}-u^{i}\big{)}\dfrac{\partial\sigma_{a}^{j}}{\partial
u^{i}}=\sigma_{a}^{i}-\sigma_{a}^{j}.$ (8.4)

If $u^{i}\neq u^{j}$ for $i\neq j$, then

$\displaystyle\dfrac{\partial\sigma_{a}^{j}}{\partial
u^{i}}=\dfrac{\sigma_{a}^{i}-\sigma_{a}^{j}}{u^{j}-u^{i}},\qquad\det\big{[}\sigma_{a}^{i}\big{]}\neq
0,\qquad a=0,1,\ldots,n-1,\qquad i=1,\ldots,n.$

It follows that for any coordinate system $(q^{i})$,

$\displaystyle\partial_{i}\sigma_{a}^{j}=\partial_{i}u^{h}\;\dfrac{\partial\sigma_{a}^{j}}{\partial
u^{h}}=\sum_{h\neq j}\partial_{i}u^{h}\;\sigma_{a-1}^{hj}=\sum_{h\neq
j}\partial_{i}u^{h}\;\dfrac{\sigma_{a}^{h}-\sigma_{a}^{j}}{u^{j}-u^{h}}.$
(8.5)

######  Theorem 8.1.

Let $\boldsymbol{L}$ be a symmetric $2$-tensor with eigenvalues $(u^{i})$. The
tensors
$(\boldsymbol{K}_{a})=(\boldsymbol{K}_{0},\boldsymbol{K}_{1},\ldots,\boldsymbol{K}_{n-1})$
defined by

$\displaystyle\boldsymbol{K}_{0}=\boldsymbol{G},\qquad\boldsymbol{K}_{a}=\tfrac{1}{a}\operatorname{tr}(\boldsymbol{K}_{a-1}\boldsymbol{L})\boldsymbol{G}-\boldsymbol{K}_{a-1}\boldsymbol{L},\qquad{{a>0}},$
(8.6)

or by

$\displaystyle\boldsymbol{K}_{a}=\sum_{k=0}^{a}(-1)^{k}\sigma_{a-k}\boldsymbol{L}^{k},\qquad\boldsymbol{K}_{a}=\sigma_{a}\boldsymbol{G}-\boldsymbol{K}_{a-1}\boldsymbol{L},\qquad\boldsymbol{K}_{-1}=\boldsymbol{0},$
(8.7)

form a basis of a KS-space if and only if $\boldsymbol{L}$ is an L-tensor.

###### Proof.

(i) Assume that $(\boldsymbol{K}_{a})$ defined by (8.6) is a basis of a KS-
space. Then they are linearly independent and there are orthogonal (separable)
coordinates in which these tensors as well as $\boldsymbol{L}$ are
diagonalised and equations

$\displaystyle\partial_{i}\rho_{a}^{j}=\big{(}\rho_{a}^{i}-\rho_{a}^{j}\big{)}\partial_{i}\ln
g^{jj},\qquad\det\big{[}\rho_{a}^{i}\big{]}\neq 0,$ (8.8)

hold, being $\rho_{a}^{i}$ the eigenvalues of $\boldsymbol{K}_{a}$. Due to
(8.6), these eigenvalues fulfill the recurrence relation
$\rho_{a}^{i}=\tfrac{1}{a}\sum_{k}\rho_{a-1}^{k}u^{k}-\rho_{a-1}^{i}u^{i}$. On
the other hand, from the first equations (8.3) and (8.2) we get
$\tfrac{1}{a}\sum_{k}\sigma_{a-1}^{k}u^{k}-\sigma_{a-1}^{i}u^{i}=\sigma_{a}-\sigma_{a-1}^{i}u^{i}=\sigma_{a}^{i}$.
This shows that

$\displaystyle\rho_{a}^{i}=\sigma_{a}^{i}(\underline{u}).$

It follows that: (I) Due to the second equations (8.3) and (8.8), the
eigenvalues $u^{i}$ are simple. (II) Due to the first equation (8.2), the
definition (8.6) implies the alternative definitions (8.7). (III) Due to the
first equation (8.8), $\partial_{i}\rho_{a}^{i}=0$ thus, $0=\sum\limits_{h\neq
i}\partial_{i}u^{h}\sigma_{a-1}^{hi}$. Let us consider the case $i=1$. We get
the linear homogeneous system of $n-1$ equations

$\displaystyle\sum_{h>1}\partial_{1}u^{h}\sigma_{a-1}^{h1}=0,\qquad
a=1,\ldots,n-1,$ (8.9)

in the $n-1$ unknown functions $\partial_{1}u^{h}$, with $h=2,\ldots,n$. We
can put $\sigma_{a-1}^{h1}=\tilde{\sigma}_{b}^{h}$, where
$\tilde{\sigma}_{b}^{h}$, $b=0,\ldots,n-2$, are the symmetric polynomials in
the $n-1$ variables $(u^{2},\ldots,u^{n})$. In analogy with the third equation
(8.4) we have
$\det[\tilde{\sigma}_{b}^{h}]=\prod\limits_{j>i>1}(u^{i}-u^{j})$, thus
$\det[\tilde{\sigma}_{b}^{h}]=\det[\sigma_{a-1}^{h1}]\neq 0$. It follows from
(8.9) that $\partial_{1}u^{h}=0$ for all $h>1$. In a similar way we prove that
$\partial_{i}u^{h}=0$ for all $h\neq i$. This shows that
$\boldsymbol{H}(\boldsymbol{L})=0$. Finally, from
$\boldsymbol{K}_{1}=\text{tr}(\boldsymbol{K}_{1}\boldsymbol{L})-\boldsymbol{L}$
we get

$\displaystyle\boldsymbol{L}\doteq\frac{1}{n-1}\operatorname{tr}(\boldsymbol{K}_{1})\boldsymbol{G}-\boldsymbol{K}_{1}.$
(8.10)

Being $\boldsymbol{K}_{1}$ a KT, $\boldsymbol{L}$ is a CKT. (ii) Conversely,
assume that $\boldsymbol{L}$ is an L-tensor. In coordinates associated with
$\boldsymbol{L}$ we have $\partial_{i}u^{h}=0$ for all $h\neq i$, and
moreover, $0=(u^{i}-u^{j})\partial_{i}\ln g^{jj}+\partial_{i}u^{i}$. By (8.5)
we get

$\displaystyle\partial_{i}\sigma_{a}^{j}=\partial_{i}u^{i}\dfrac{\sigma_{a}^{i}-\sigma_{a}^{j}}{u^{j}-u^{i}}=\big{(}\sigma_{a}^{i}-\sigma^{j}_{a}\big{)}\partial_{i}\ln
g^{jj}.$

Being $\rho_{a}^{i}=\sigma_{a}^{i}$, this shows that the tensors
$\boldsymbol{K}_{a}$ are Killing tensors. They are pointwise independent due
to (8.3). ∎

We call L-system any separable orthogonal system whose KS-space is generated
by an L-tensor according to Theorem 8.1. We call L-web any orthogonal web
corresponding to an L-system333L-systems have been considered in [17, 18]. See
also the extensive classification [9] of special symmetric two-tensors
(including L-tensors) with application to the theory of the equivalent
dynamical systems and to the theory of cofactor and bi-cofactor systems..

######  Remarks 8.2.

  1. (i)

All the Killing tensors $\boldsymbol{K}_{a}$ ($a\neq 0$) have simple
eigenvalues. The sequence (8.6) was suggested by the analysis [4] of the
planar inertia tensor $\boldsymbol{L}$ of an asymmetric massive body in the
Euclidean $n$-space. The tensor $\boldsymbol{K}_{1}$ is the corresponding
inertia tensor. Formula (8.7) shows that $\boldsymbol{K}_{a}=0$ for $a>n-1$,
since for $a=n$ the right-hand side vanishes due to the Hamilton–Cayley
theorem. Formula (8.6) is more effective than (8.7) since it does dot require
the knowledge of the eigenvalues of $\boldsymbol{L}$.

  2. (ii)

Within a completely different context, a sequence like (8.6) is considered by
Schouten [49, p. 30] generated by a matrix $\boldsymbol{P}$, as an effective
tool for computing the eigenvectors of a matrix $\boldsymbol{P}$ without
solving systems of linear equations (when the eigenvalues are known and
simple). Such a method was firstly introduced by Fettis [28] and Souriau [51].

  3. (iii)

We observe with Schouten [49] that the (1,1) tensor
$\boldsymbol{Q}(x)=\operatorname{cof}(\boldsymbol{L}-x\boldsymbol{G})$ is
polynomial of degree $n-1$ in $x$, whose coefficients, up to the sign, are the
tensors $\boldsymbol{K}_{a}$ defined in (8.7). We recall that the cofactor
$\tilde{\boldsymbol{A}}=\operatorname{cof}(\boldsymbol{A})$ of
$\boldsymbol{A}$ is defined by
$\boldsymbol{A}\tilde{\boldsymbol{A}}=\tilde{\boldsymbol{A}}\boldsymbol{A}=\det(\boldsymbol{A})\boldsymbol{G}$.
Hence, the Stäckel systems of the kind considered in the last two theorems are
just the so-called cofactor systems (cf. [16]). In fact, they should be called
Levi-Civita systems (so that “L-systems” is a good notation) since the
separable metric (7.3) (but not the tensor $\boldsymbol{L}$) appears for the
first time in [36], where it is also shown that for such a metric the function

$\displaystyle F(\underline{q},\underline{\dot{q}},c)\doteq\prod_{j\neq
i}\big{(}u^{j}+c\big{)}\big{|}u^{j}-u^{i}\big{|}\big{(}\dot{q}^{i}\big{)}^{2}$

is a first integral of the geodesics for all values of the parameter $c$.
Since $F$ is a polynomial in $c$ of degree $n-1$, its coefficients gives rise
to $n$ first integrals. These first integrals coincide, up to the sign and
after the Legendre transformation, with the first integrals
$P(\boldsymbol{K}_{a})$.

  4. (iv)

Due to item (v) of Theorem 7.3, a necessary condition for a Stäckel system to
be an L-system is the Robertson condition:
$[\\![\boldsymbol{K},\boldsymbol{R}]\\!]=0$ for a characteristic tensor
$\boldsymbol{K}$ (thus, for all elements of the KS-space).

A criterion for testing if a Stäckel system is an L-system is the following.

######  Theorem 8.3.

A Stäckel system is an L-system if and only if in the corresponding KS-space
there exists a characteristic tensor $\boldsymbol{K}_{1}$ such that the tensor
$\boldsymbol{L}$ defined by (8.10) is torsionless.

###### Proof.

(i) Assume that there exists a ChKT $\boldsymbol{K}_{1}$ such that the tensor
$\boldsymbol{L}$ defined by (8.10) is torsionless. Any tensor of the kind
$f\boldsymbol{G}+\boldsymbol{K}$ is a CKT if $\boldsymbol{K}$ is a KT. Since
$\boldsymbol{K}_{1}$ has simple eigenvalues, also $\boldsymbol{L}$ has simple
eigenvalues. Then $\boldsymbol{L}$ is an L-tensor. The tensors
$\boldsymbol{K}_{a}$ constructed by applying (8.6) form a KS-space $\cal
K_{*}$ which has $\boldsymbol{K}_{1}$ in common with the original KS-space
$\cal K$. Thus, $\cal K_{*}=\cal K$ since two KS-spaces with a ChKT in common
coincide. (ii) The converse is obvious. ∎

About the uniqueness of an L-tensor generating an L-system, it can be proved
that

######  Theorem 8.4.

Two L-tensors $\boldsymbol{L}$ and $\tilde{\boldsymbol{L}}$ generates the same
L-system if and only if
$\tilde{\boldsymbol{L}}=a\boldsymbol{L}+b\boldsymbol{G}$, $a,b\in\mathbb{R}$,
$a\neq 0$.

In accordance with the remarks at the end of Section 5, a basis
$(\boldsymbol{K}_{a})$ of a KS-space corresponds to the inverse of a Stäckel
matrix $\boldsymbol{S}^{-1}=\big{[}\varphi_{(a)}^{i}\big{]}$,
$\varphi_{(a)}^{i}=K_{a}^{ii}$. In the sequence (8.6) we have
$\boldsymbol{K}_{0}=\boldsymbol{G}$, then $\varphi_{(0)}^{i}=g^{ii}$ and
$\varphi_{(a)}^{i}=K_{a}^{ii}=\sigma_{a}^{i}g^{ii}$. By using the formulas
concerning the elementary symmetric function at the beginning of Section 8 it
can be proved that

######  Theorem 8.5.

For an L-system the Stäckel matrix associated with the basis (8.6),
$\boldsymbol{S}=\big{[}\varphi_{i}^{(a)}\big{]}$, assuming
$u^{1}<u^{2}<\cdots<u^{n}$, is of alternating Vandermonde type,

$\displaystyle\varphi_{i}^{(a)}=(-1)^{n-a+i}(u^{i})^{n-a-1},\qquad
a=0,\ldots,n-1,\qquad i=1,\ldots,n\quad\hbox{\rm index of row.}$ (8.11)

For $n=3$ we have ($a$ index of row)

$\displaystyle\big{[}\varphi_{(a)}^{i}\big{]}=\begin{bmatrix}g^{11}&g^{22}&g^{33}\\\\[3.0pt]
(u^{2}+u^{3})g^{11}&(u^{3}+u^{1})g^{22}&(u^{1}+u^{2})g^{33}\\\\[3.0pt]
u^{2}u^{3}g^{11}&u^{3}u^{1}g^{22}&u^{1}u^{2}g^{33}\end{bmatrix},$
$\displaystyle g^{11}=\dfrac{1}{(u^{1}-u^{2})(u^{1}-u^{3})},\qquad
g^{22}=\dfrac{1}{(u^{2}-u^{3})(u^{1}-u^{2})},\qquad
g^{33}=\dfrac{1}{(u^{3}-u^{2})(u^{3}-u^{1})}.$

In accordance with Theorem 8.1, the inverse matrix is

$\displaystyle\boldsymbol{S}=[\varphi_{i}^{(a)}]=\begin{bmatrix}(u^{1})^{2}&-u^{1}&1\\\\[6.0pt]
-(u^{2})^{2}&u^{2}&-1\\\\[6.0pt] (u^{3})^{2}&-u^{3}&1\cr\end{bmatrix}.$

######  Remark 8.6.

If we multiply each row $\varphi_{i}^{(a)}$ of a Stäckel matrix by a function
$f_{i}(q^{i})\neq 0$, then we get a new Stäckel matrix
$\tilde{\varphi}_{i}^{(a)}=f_{i}\varphi_{i}^{(a)}$ whose inverse
$\tilde{\varphi}^{i}_{(a)}$ defines a new basis of the same KS-space. By
multiplying the lines of (8.11) by $\pm 1$ in a suitable way and by changing
$\boldsymbol{L}$ in $-\boldsymbol{L}$, we can get a Stäckel matrix which is of
the Vandermonde type (see the case $n=3$, for instance).

##  9 The functional independence of the eigenvalues of an L-tensor

The results of Section 8 hold without any assumption on the functional
independence of the eigenvalues $(u^{i})$ of $\boldsymbol{L}$. Some of them
may be constant. The only essential assumption is that they are pairwise and
pointwise distinct. The following theorems deal with this matter.

######  Theorem 9.1.

If a torsionless CKT has functionally independent eigenvalues $(u^{i})$, then
it is an L-tensor $($its eigenvalues are pointwise simple$)$.

###### Proof.

In this case the eigenvalues define local orthogonal coordinates $q^{i}=u^{i}$
in which the tensor is diagonalised. We can apply Theorem 6.2. Equations (6.2)
become $\delta_{i}^{k}=(u^{i}-u^{k})\partial_{i}\ln g^{kk}+1$. For $i\neq k$
we have $u^{i}\neq u^{k}$. ∎

Note that in this case $\boldsymbol{L}$ is a special type of L-tensor (the
type considered by Crampin [22, 23]).

######  Theorem 9.2.

Let $\boldsymbol{L}$ be an L-tensor. $(i)$ The eigenvalues $(u^{i})$ of
$\boldsymbol{L}$ are independent functions $($i.e., they define locally an
orthogonal coordinate system$)$ if and only if $\boldsymbol{L}$ is not
invariant with respect to a Killing vector $\boldsymbol{X}$. $(ii)$ If there
exists a Killing vector $\boldsymbol{X}$ such that
$[\boldsymbol{X},\boldsymbol{L}]=0$, then $\boldsymbol{X}$ is a linear
combination $($with constant coefficients$)$ of Killing vectors in involution
which are eigenvectors of $\boldsymbol{L}$.

In case (i) we have no symmetry of the separable web generated by the
eigenvectors of $\boldsymbol{L}$. This means that all the structures
associated with $\boldsymbol{L}$ are not invariant with respect to groups of
isometries.

###### Proof.

The eigenvalues of a (1,1)-tensor $\boldsymbol{L}$ are functionally
independent, and thus they define a coordinate system, if and only if
$\det[\partial_{i}u^{j}]\neq 0$. For a torsionless tensor with simple
eigenvalues we have, with respect to the associated coordinates,
$\partial_{i}u^{j}=0$ for $i\neq j$. Hence, the eigenvalues are independent
functions if and only $\partial_{i}u^{i}\neq 0$ for all indices. (i) If an
eigenvalue $u^{i}$ of $\boldsymbol{L}$ is constant, $\partial_{i}u^{i}=0$,
then from the expression (7.3) of the metric we see that the corresponding
(normalised) coordinate $q^{i}$ is ignorable. This means that
$\partial/\partial q^{i}$ is a Killing vector which leaves invariant
$\boldsymbol{L}$. Conversely, let us assume that there is a Killing vector
$\boldsymbol{X}$ such that $[\boldsymbol{X},\boldsymbol{L}]=0$. These two
assumptions on $\boldsymbol{X}$ are equivalent to
$\\{P_{\boldsymbol{X}},P_{\boldsymbol{G}}\\}=0$ and
$\\{P_{\boldsymbol{X}},P_{\boldsymbol{L}}\\}=0$. In an orthogonal coordinate
system $(q^{i})$ associated with $\boldsymbol{L}$, these two equations read
$\\{X^{i}p_{i},g^{jj}p_{j}^{2}\\}=0$ and
$\\{X^{i}p_{i},u^{j}g^{jj}p_{j}^{2}\\}=0$, respectively. Since they are
algebraic equations in $\underline{p}$, to be satisfied for any value of these
variables,they are equivalent to

$\displaystyle\sum_{i}X^{i}\partial_{i}g^{kk}-2g^{kk}\partial_{k}X^{k}=0,\qquad
g^{jj}\partial_{j}X^{k}+g^{kk}\partial_{k}X^{j},\qquad j\neq k,$
$\displaystyle\sum_{i}X^{i}\partial_{i}\big{(}u^{k}g^{kk}\big{)}-2u^{k}g^{kk}\partial_{k}X^{k}=0,\qquad
u^{j}g^{jj}\partial_{j}X^{k}+u^{k}g^{kk}\partial_{k}X^{j},\qquad j\neq k.$

The first two equations characterise the Killing vectors in orthogonal
coordinates. The second and the fourth equations imply
$(u^{j}-u^{k})g^{jj}\partial_{j}X^{k}=0$ for $j\neq k$. Since $u^{j}\neq
u^{k}$ we conclude that $\partial_{j}X^{k}=0$ for $j\neq k$, which means that
$X^{i}=X^{i}(q^{i})$. Since $\partial_{i}u^{k}=0$ for $i\neq k$, from the
third equation it follows that
$X^{k}\partial_{k}u^{k}g^{kk}+u^{k}\sum_{i}X^{i}\partial_{i}g^{kk}-2u^{k}g^{kk}\partial_{k}X^{k}=0$.
Due to the first equation (8.9), this last equation implies
$X^{k}\partial_{k}u^{k}=0$ (no summation over the index $k$). Up to a
reordering of the coordinates, let us assume that $X^{a}=0$ and
$X^{\alpha}\neq 0$ for $a=1,\ldots,m$ and $\alpha=m+1,\ldots,n$. From the last
equation it follows that $u^{\alpha}={\rm const}$ and
$\boldsymbol{X}=X^{\alpha}\partial_{\alpha}$. At the beginning of this proof
we have seen that if $u^{\alpha}={\rm const}$, then $q^{\alpha}$ are ignorable
coordinates (we always assume that the coordinates $(q^{i})$ are normalised so
that the metric assumes the form (7.3)). Thus, $\partial_{\alpha}$ are Killing
vectors in involution and eigenvectors of $\boldsymbol{L}$. Since
$\boldsymbol{X}=X^{\alpha}\partial_{\alpha}$ is a Killing vector, the
components $X^{a}$ must be constant. ∎

Assume that the eigenvalues $u^{\alpha}$ ($u^{\alpha}=m+1,\ldots,n$) of an
L-tensor are constant and the remaining $(u^{a})$ ($a=1,\ldots,m$) are
independent functions. Then we can choose associated orthogonal coordinates
$(q^{i})=(q^{a},q^{\alpha})$ such that $q^{a}=u^{a}$ and $q^{\alpha}$ are
ignorable. From (7.1) it follows that

$\displaystyle\big{(}u^{a}-u^{b}\big{)}\partial_{a}\ln
g^{bb}+1=0,\qquad\big{(}u^{a}-u^{\alpha}\big{)}\partial_{a}\ln
g^{\alpha\alpha}+1=0,$

being the remaining equations identically satisfied. Thus, we are faced with
three cases:

  1. (I)

$m=0$, all $u^{i}={\rm const}$, i.e., all $q^{i}$ are ignorable: the manifold
$Q$ is locally flat, the coordinates $(q^{i})$ are orthogonal Cartesian
coordinates, $g^{ii}={\rm const}$ and $\boldsymbol{L}$ is a constant tensor.

  2. (II)

$0<m<n$: in this case $g^{\alpha\alpha}\neq{\rm const}$ due to equation
$(u^{a}-u^{\alpha})\partial_{a}\ln g^{\alpha\alpha}+1=0$. Condition
$g^{\alpha\alpha}\neq{\rm const}$ means that the Killing vectors
$\boldsymbol{X}_{\alpha}=\partial_{\alpha}$ are not translations [25, Section
52].

  3. (III)

$m=n$, all eigenvalues are independent: this is the case examined in Theorem
9.2.

Taking into account the proof of the preceding theorem, cases (I) and (II)
shows that:

######  Theorem 9.3.

Let $\boldsymbol{L}$ be an L-tensor. $(i)$ If $\boldsymbol{L}$ has all
constant eigenvalues, then the manifold $Q$ is locally flat and
$\boldsymbol{L}={\rm const}$ $($in the sense that all its components in
Cartesian coordinates are constant$)$. $(ii)$ If $\boldsymbol{L}$ is invariant
with respect to $m<n$ Killing vectors, then these vectors are not
translations.

##  10 L-potentials

In this section we propose a few remarks on the potential functions associated
with an L-system. For further approaches to this matter we refer the reader to
[16, 30]. By virtue of (1.3) and (7.3) we have

######  Theorem 10.1.

A potential $V$ is separable in an L-system if and only if, with respect to
normal coordinates $(q^{i})$ associated with $\boldsymbol{L}$,

$\displaystyle\partial_{i}\partial_{j}V=\dfrac{1}{u^{j}-u^{i}}(\partial_{j}V-\partial_{i}V),\qquad\partial_{i}=\dfrac{\partial}{\partial
q^{i}}.$

Note that in this theorem (as well as in the following) the eigenvalues
$u^{i}$ of $\boldsymbol{L}$ may not be independent functions.

######  Theorem 10.2.

Let $(\boldsymbol{K}_{a})$ be the basis of the KS-space generated by an
L-tensor $\boldsymbol{L}$ according to formula (8.6). Then the functions
$V_{a}(\underline{u})\doteq\sigma_{a+1}(\underline{u})$ $(a=0,1,\ldots,n-1)$
fulfill equations $dV_{a}=\boldsymbol{K}_{a}\,dV$, with
$V=V_{0}=\operatorname{tr}(\boldsymbol{L})=\sum_{i}u^{i}$, and the functions
$H_{a}=\tfrac{1}{2}P_{\boldsymbol{K}_{a}}+V_{a}$ are first integrals in
involution.

###### Proof.

$\boldsymbol{K}_{a}\,dV=\sigma_{a}^{i}\partial_{i}V\,dq^{i}=\sigma_{a}^{i}\,dq^{i}=\partial_{i}\sigma_{a+1}\,dq^{i}=dV_{a}$.
Apply (5.1). ∎

##  11 L-pencils

There are Stäckel webs which are not L-webs. A necessary condition is the
Robertson condition. However, also in manifolds where this condition is
identically satisfied, for instance in constant curvature spaces, there are
Stäckel systems which are not generated by an L-tensor. For instance, due to
Theorem 9.3(ii), in a $\mathbb{E}_{n}$ all translational webs (except the
Cartesian web) are not L-webs. As remarked above, also the spherical-conical
webs are not L-webs. This last case has suggested [4] the introduction of a
linear combination444See also [10].

$\displaystyle\boldsymbol{L}(m)=\boldsymbol{L}_{0}+m\boldsymbol{L}_{1},$

which is an L-tensor for all values the parameter $m\in\mathbb{R}$. We call
L-pencil such an object.

######  Theorem 11.1.

Let $\boldsymbol{L}=\boldsymbol{L}_{0}+m\boldsymbol{L}_{1}$ be an L-pencil.
Then:

  1. $(i)$

$\boldsymbol{L}_{0}$ has simple eigenvalues.

  2. $(ii)$

$\boldsymbol{L}$ is a CKT for all $m$ if and only if $\boldsymbol{L}_{0}$ and
$\boldsymbol{L}_{1}$ are CKT’s.

  3. $(iii)$

The condition $\boldsymbol{H}(\boldsymbol{L})=0$ is equivalent to

$\displaystyle\boldsymbol{H}(\boldsymbol{L}_{0})=\boldsymbol{H}(\boldsymbol{L}_{1})=0,\qquad\underset{0}{L}{\phantom{|}}_{[i}^{h}\partial_{|h|}\underset{1}{L}{\phantom{|}}_{j]}^{k}-\underset{0}{L}{\phantom{|}}_{h}^{k}\partial_{[i}\underset{1}{L}{\phantom{|}}_{j]}^{h}+\underset{1}{L}{\phantom{|}}_{[i}^{h}\partial_{|h|}\underset{0}{L}{\phantom{|}}_{j]}^{k}-\underset{1}{L}{\phantom{|}}_{h}^{k}\partial_{[i}\underset{0}{L}{\phantom{|}}_{j]}^{h}=0.$

  4. $(iv)$

The Ricci tensor $\boldsymbol{R}$ commutes with both $\boldsymbol{L}_{0}$ and
$\boldsymbol{L}_{1}$.

The proof is straightforward. Let us apply the iterative formula (8.6) and
Theorem 8.1 to the tensor $\boldsymbol{L}(m)$. Since $\boldsymbol{L}$ is
polynomial of degree 1 in $m$, each tensor $\boldsymbol{K}_{a}(m)$ is at most
of degree $a$:
$\boldsymbol{K}_{a}(m)=\boldsymbol{H}_{a}m^{a}+\boldsymbol{H}_{a-1}m^{a-1}+\cdots$.

######  Theorem 11.2.

$(i)$ The tensors $\boldsymbol{K}_{a}(0)$ form a KS-space. $(ii)$ The tensors
$\boldsymbol{H}_{a}$, given by the coefficients of maximal degree of
$\boldsymbol{K}_{a}(m)$, if independent, form a KS-space. $(iii)$ The Stäckel
systems generated by these two KS-spaces satisfy the Robertson condition.

Also for this statement we omit the proof. We remark that there are cases in
which all the $\boldsymbol{K}_{a}$ are of degree 1, as in the following
example.

######  Example 11.3 ([4]).

In $\mathbb{E}_{n}$ we consider the tensor
$\boldsymbol{L}(m)=\boldsymbol{L}_{0}+m\boldsymbol{r}\otimes\boldsymbol{r}$.
It represents (for $m\neq 0$) the planar moment of inertia of a massive body
with total mass $m$ and center of mass at the origin. $\boldsymbol{L}_{0}$ is
a constant symmetric tensor (hence, a KT) with simple eigenvalues $(a^{i})$.
$\boldsymbol{L}_{1}=\boldsymbol{r}\otimes\boldsymbol{r}$ is a CKT whose
eigenvalues are all zero except one ($={}r^{2}$). It can be proved that: (i)
$\boldsymbol{L}$ is an L-pencil; (ii) the tensors $\boldsymbol{K}_{a}(0)$ form
a KS-space corresponding to Cartesian coordinates; (iii)
$\boldsymbol{K}_{a}(m)$ are all of degree 1 in $m$,
$\boldsymbol{K}_{a}(m)=\boldsymbol{K}_{a}(0)+m\boldsymbol{H}_{a}$; (iv) the
tensors $\boldsymbol{H}_{a}$ form a KS-space corresponding to the conical
spherical coordinates.

The above results stimulate investigations about the notion of an L-pencil,
also in relation with recent studies on the same concept and the notion of
cofactor pair system (see, e.g., [38, 39, 47, 48]). A possible generalisation
is a multi-pencil of the type
$\boldsymbol{L}=\boldsymbol{L}_{0}+m^{i}\boldsymbol{L}_{i}$ with
$(m^{i})\in\mathbb{R}^{k}$ or $\boldsymbol{L}=m^{i}\boldsymbol{L}_{i}$ with
$(m^{i})\neq\boldsymbol{0}$. Basic examples are, in the Euclidean $n$-space,
the case
$\boldsymbol{L}=\boldsymbol{L}_{0}+m\boldsymbol{r}\otimes\boldsymbol{r}+w\boldsymbol{c}\odot\boldsymbol{r}$
where $\boldsymbol{c}$ is a constant unit vector, see (4.2). For brevity, we
do not examine here which Stäckel webs in $\mathbb{E}_{n}$ (or in
$\mathbb{S}_{n}$, $\mathbb{H}_{n}$) are generated by an L-tensor or by an
L-pencil.

##  12 The Riemannian background of the separation

Also for a better understanding of the orthogonal separation, it is necessary
to study the Levi-Civita equations without any a priori assumptions on the
separable coordinates $\underline{q}$. The geometrical meaning of these
equations has been firstly investigated by Kalnins and Miller [32, 33] and in
[2] (see also [3, 7]). A result of these investigations is the following.

######  Theorem 12.1 ([8]).

The Hamilton–Jacobi equation $G+V=E$ is separable if and only if there exists
a characteristic Killing pair $(D,\boldsymbol{K})$ such that

$\displaystyle DV=0,\qquad d(\boldsymbol{K}\,dV)=0.$ (12.1)

A characteristic Killing pair $(D,\boldsymbol{K})$ is made of a
$r$-dimensional linear space $D$ of commuting Killing vectors and a
$D$-invariant Killing tensor $\boldsymbol{K}$ having $m=n-r$ normal
eigenvectors orthogonal to $D$ and corresponding to distinct eigenvalues (note
that these eigenvalues may not be simple). Since the elements of $D$ are in
involution, they generate $r$-dimensional orbits which are locally flat
submanifolds. It can be shown that if $(D,\boldsymbol{K})$ is a characteristic
Killing pair, then $D$ is normal, in the sense that the distribution
orthogonal to its elements is completely integrable. This means that there is
a foliation of $m$-dimensional manifolds orthogonal to the orbits of $D$. The
leaves of this foliation are isometric Riemannian manifolds (the isometries
being generated by the orbits of $D$, assuming that they intersect the orbits
in only one point).

With a characteristic Killing pair we associate standard separable coordinates
$(q^{i})=(q^{a},q^{\alpha})$ ($a=1,\ldots,m$, $\alpha=m+1,\ldots,n$) defined
in this way: (i) $dq^{a}$ are eigenforms of $\boldsymbol{K}$ orthogonal to $D$
(or, equivalently, the orbits of $D$ are defined by equations $q^{a}={\rm
const}$); (ii) the $(q^{\alpha})$ are the affine parameters, with zero value
on an arbitrary $m$-dimensional orthogonal section $\cal Z$ of the orbits of
$D$, of Killing vectors $\boldsymbol{X}_{\alpha}$ forming a local basis of
$D$. It follows that: (I) $\partial/\partial
q^{\alpha}=\boldsymbol{X}_{\alpha}$ and the coordinates $(q^{\alpha})$ are
ignorable; (II) in these coordinates the contravariant metric tensor
components assume the standard form

$\displaystyle\big{[}g^{ij}\big{]}=\begin{bmatrix}\big{[}g^{aa}\big{]}&[0]\\\\[3.0pt]
[0]&\big{[}g^{\alpha\beta}\big{]}\cr\end{bmatrix},$ (12.2)

where $\big{[}g^{aa}\big{]}$ is a diagonal $m\times m$ matrix and
$\big{[}g^{\alpha\beta}\big{]}$ is a $r\times r$ matrix, with $m=n-r$. Here we
denote by $[0]$ zero-matrices of proper dimensions. The rather long proof of
this theorem is based on the following

######  Theorem 12.2 ([2]).

In an equivalence class of separable coordinates there exist coordinates in
which the metric assume the standard form (12.2).

Two separable systems are called equivalent if (in the intersection of their
domains of definition) the corresponding separated solutions of the
Hamilton–Jacobi equation generate the same Lagrangian foliation of $T^{*}Q$.
The geometrical representation of an equivalent class is given by a separable
web: it is a pair $(D,{\cal S}^{a})=(D,({\cal S}^{1},\ldots,{\cal S}^{m}))$,
where $D$ is a normal $r$-dimensional space of commuting KT’s and ${(\cal
S}^{a})$ is a family of $m$ orthogonal foliations of submanifolds of
codimension 1, all invariant with respect to $D$ (it follows that the orbits
of $D$ are the complete intersections of the $({\cal S}^{a})$). Moreover, the
submanifolds $({\cal S}^{a})$ are orthogonal to $m$ eigenvectors of a
$D$-invariant KT $\boldsymbol{K}$ with distinct (but not necessarily simple)
eigenvalues (so that $(D,\boldsymbol{K})$ is a characteristic Killing pair).
It must be observed that the quotient set of the orbits of $D$ is a
$m$-dimensional manifold with a naturally induced orthogonal separable metric
$(g^{aa})$ (it is isomorphic to any $m$-manifold orthogonal to $D$).

In this description of the separation we include the extreme cases: (I) $m=n$,
$r=0$, which corresponds to the an orthogonal separation; (II) $m=0$, $r=n$,
which corresponds to the separation in Cartesian coordinates (in this case the
manifold is flat). In [13] it is proved that

######  Theorem 12.3.

A characteristic Killing pair $(D,\boldsymbol{K})$ generates a $m$-dimensional
space ${\cal K}_{m}$ of Killing tensors with the following properties: $(i)$
they are $D$-invariant and have $m$ eigenvectors in common orthogonal to $D$;
$(ii)$ they are in involution; $(iii)$ if the characteristic equations (12.1)
of a separable potential $V$ are satisfied, then they are satisfied by all
elements of ${\cal K}_{m}$.

######  Theorem 12.4.

In a space ${\cal K}_{m}$ having properties $(i)$ and $(ii)$, where $D$ is a
$n-m$-space of Killing vectors in involution, all the common eigenvectors are
normal.

######  Theorem 12.5.

The functions $H_{a}=\tfrac{1}{2}P_{\boldsymbol{K}_{a}}+V_{a}$ and
$P_{\boldsymbol{X}_{\alpha}}=X_{\alpha}^{i}p_{i}$, where
$dV_{a}=\boldsymbol{K}_{a}\,dV$ and $(\boldsymbol{X}_{\alpha})$ is a basis of
$D$, form a complete system of integrals in involution.

Theorems 12.2 and 12.3 show that if $D$ admits an orthogonal basis,
$\boldsymbol{X}_{\alpha}{\mathbf{\cdot}}\boldsymbol{X}_{\beta}=0$ for
$\alpha\neq\beta$, then the standard coordinates are orthogonal. This means
that the separation occurs in orthogonal coordinates. It can be proved by a
coordinate-independent method that

######  Theorem 12.6 ([5]).

On a manifold with constant curvature any normal space $D$ of Killing vectors
in involution has an orthogonal basis.

As a consequence,

######  Theorem 12.7.

On a manifold of constant curvature the geodesic separation always occurs in
orthogonal coordinates.

This important property was discovered by Kalnins and Miller [34, 35] (see
also [31]), within a coordinate-dependent approach. This theorem can be
extended to a natural Hamiltonian $H=G+V$.555The study of the Riemannian
background of the separation has been extended to a Hamiltonian with scalar
and vector potential in [12].

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