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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 21. Выпуск 2.
УДК 514.7, 514.8 DOI 10.22405/2226-8383-2020-21-2-275-289
Квантовая интегрируемость операторов Бельтрами — Лапласа проективно эквивалентных метрик произвольных сигнатур.
В. С. Матвеев
Матвеев Владимир Сергеевич — кандидат физико-математических наук, профессор, Иен-ский университет имени Фридриха Шиллера (г. Иена, Германия). e-mail: Vladimir. matveev <3uni-jena. de
Аннотация
Мы обобщаем результат [31] на все сигнатуры.
Ключевые слова: интегрируемые системы, тензоры Киллинга, квантовые интегрируемые системы, коммутирующие операторы, квантование по Кантеру, проективно эквивалентные метрики, геодезически эквивалентные метрики, разделение переменных, нормальные формы, геометрическая теория УрЧП, голоморфно-проективно эквивалентные метрики
Библиография: 51 название. Для цитирования:
В. С. Матвеев. Квантовая интегрируемость операторов Бельтрами — Лапласа проективно эквивалентных метрик произвольных сигнатур // Чебышевский сборник, 2020, т. 21, вып. 2, с. 275-289.
CHEBYSHEVSKII SBORNIK Vol. 21. No. 2.
UDC 514.7, 514.8 DOI 10.22405/2226-8383-2020-21-2-275-289
Quantum integrability for the Beltrami^Laplace operators of projectively equivalent metrics of arbitrary signatures
V. S. Matveev
Matveev Vladimir Sergeevich — candidate of physical and mathematical Sciences, Professor, Friedrich-Schiller-Universitât Jena, (Jena, Germany). e-mail: vladimir. matveev <§uni-jena. de
Abstract
We generalize the result of [31] to all signatures.
Keywords: integrable systems, Killing tensors, quantum integrable systems, Carter quantisation, commutative operators, projectively equivalent metrics, geodesically equivalent metrics, separation of variables, normal forms, geometric theory of PDE, c-projectively equivalent metrics
Bibliography: 51 titles.
For citation:
V. S. Matveev, 2020, "Quantum integrabilitv for the Beltrami-Laplace operators of projectivelv equivalent metrics of arbitrary signatures" , Chebyshevskii sbornik, vol. 21, no. 2, pp. 275-289.
Dedicated to Anatoly Timofeevich Fomenko on his 75th birthday.
1. Introduction
Let M be a smooth manifold of dimension n ^ 2. We say that two metrics g and g on this manifold are projectively equivalent, if each g-geodesic, after a proper reparameterization, is a g-geodesic. Theory of projectively equivalent metrics is a classical topic in differential geometry, already E. Beltrami fl] and T. Levi-Civita [26] did important contributions there. In the last two decades a group of new methods coming from integrable systems, see e.g. [27, 28, 29, 32, 33], and from Cartan geometry, see e.g. [19, 39, 44], appeared to be useful in this theory, and made it possible to solve important open problems and named conjectures, see e.g. [37, 34, 12, 40, 38].
By [27, 35] the existence of g projectively equivalent to g allows one to construct a family K^^ of Killing tensors of second degree for the metric g (we will recall the formula and the definition later, in §2.1, following later publications, e.g. [3, 34, 36]. The family K^^ is polynomial in t of degree n — 1 so it contains at most n linearly independent Killing tensors).
In this paper we answer in Theorem 1 the following natural 'quantization' question: do the corresponding second order differential operators commute?
There are of course many possible constructions of differential operators of second order by (0,2)-tensors, and, more generally, many different quantization approaches, see e.g. [9, §6]. We use the quantization procedure of B. Carter [15, Equation (6.15)] and refer to [15] and also to [2, 18] for an explanation why it is natural in many aspects. The construction is as follows: to a tensor Kij, we associate an operator
K : C™(M) ^ C™(M), K(f) = ViKijVjf. (1)
Above and everywhere in the paper V is the Levi-Civita connection of g, we sum with respect to repeating indexes and raise the indexes of K by the metric g.
Theorem 1. Assume g and g are projectively equivalent, let K(i) be the family of Killing tensors of second degree for g constructed with the help of g. Then, for any t,s € R the operators K(i), K(s) commute, that is
K WK W — K (s)K № = 0.
Note that the Beltrami-Laplace operator Aa := VVj is a linear combination of the operators of the family K(i), so all the operators K(i) commute also with Ag. In fact, in the proof we go in the opposite direction: we show first (combining [15, 18] and|22]) that the operators K(i) commute with Ag and then use this to show that the operators K(t\ Kalso commute mutually.
For Riemannian manifolds, Theorem 1 is known, it was announced in [30] and the proof appeared in [31]. The proof in the Riemannian case is based on direct calculations in the coordinates in which the metrics admit the so-called Levi-Civita normal form. These coordinates exist (locally, in a neighborhood of almost every point), if the (l,l)-tensor Glj := glsgsj is semi-simple (at almost every point). This is always the case, for example, if one of the metrics is Riemannian. The proof from [31] can be directly generalized to the pseudo-Riemannian metrics under the additional assumption that G is semi-simple.
There are (many) examples of projectively equivalent metrics such that G has nontivial Jordan blocks; in this situation the proof and ideas of [31] are not sufficient. Indeed, though also in this
case there exists a local description of projectivelv equivalent metrics [6], direct calculation of the commutators of the operators KKis a complicated task because of different combinatoric possibilities for the number and the sizes of Jordan blocks and also because the description of [6] uses a description of symmetric parallel (0,2)-tensors from [11] which is quite nontrivial. For small dimensions it is possible though to prove Theorem 1 by direct calculations, in particual in dimension 2 it was done in [4, §2.2.3].
Our proof is based on another circle of ideas, it still uses the local description of [6] but replaces local calculations by a trick which is based on quite nontrivial results of different papers. We recall the necessary results in §2.
All objects in our paper are assumed to be sufficiently smooth.
We thank C. Chanu and V. Kiosak for useful discussions.
2. Basic facts about projectively equivalent metrics and Killing tensors used in the proof
2.1. Killing tensors for projectively equivalent metrics and corresponding integrals.
construction of Killing tensors Kf^ of second degree for the metric g by using the metric g. We
Let g and g be two projectively equivalent metrics on the manifold M. Let us recall the istruction of Killing tensors Kf) of second dc consider the (l,l)-tensor L given by the formula
det(g)
тг
L3
det(g)
i
n + l
fan. (2)
Here gli is the contravariant metric dual (= inverse, i.e., glsgSj = to g.
Next, consider the family S(t), t £ R, of the (1,1)-tensors, where Id is the (l,l)-tensor corresponding to the identity endomorphism, its components in the standard tensor notation are
5%. V
S(t) := Comatrix (i Id — L). (3)
Recall that the comatrix (or the adjugate matrix) of a (l,l)-tensor is also a (l,l)-tensor. Indeed, at points where t £ Spectrum( L), it is given by
Comatrix (t Id — L) = det (t Id — L) (t Id — L)
-i
of L is everywhere dense on the real line. From the formula for the comatrix we see that the family (3) is polynomial in t of degree n — 1.
Theorem 2 (Essentially, [27]). Let g and g be projectively equivalent. Then, for every t £ R the tensor
4° := 9*S(t)5 (4)
In the coordinate-free notation the Killing tensor K(i) is given by K(t)(i, v) = g(i,S(t)v). Since L is ij-selfadjoint, S(t) is also self-adjoint so K « is symmetric with respect to the lower indexes. Recall that a (symmetric with respect to the lower indexes) tensor K^ is Killing, if
V(iKjk) = 0,
(5)
where the round brackets denote the symmetrization. In our paper we do not use this equation, but use the geometric definition which we recall now: a (0,2) symmetric tensor K = Kij is Killing, if and only if the function t ^ K(^'(t),^'(t)) is constant along every naturally parameterized g-geodesic j(t). In other words, if the function K(j'(t),j'(t)) is an integral of the geodesic flow of g. It is known, that the integrals corresponding to the Killing tensors K « constructed above commute, let us recall this statement:
Theorem 3. Let g and g be protectively equivalent and K « be the Killing tensors for g constructed by (4). Consider, for each t € R the function It : T*M ^ R given by formula
It(x,p) = Kflg^pipj. (6)
Here (x,p) = (xl, ...,xn,p\, ...,pn) are local coordinates on T*M: x1 are local coordinates on M and Pi are, for each x, the coordinates on T*.M corresponding to the basis -J^ on TXM.
Then, for any t,s € R the functions It, Is Poisson-commute with respect to the standard Poisson bracket on T*M, that is:
Edit dis. _ dh dis = 0
dpi dxl dxl dpi
i=l
In the Riemannian signature, Theorem 3 is due to [27]. In all signatures, it was independently proved in [3, 49].
2.2. Difference between connections of projectively equivalent metrics
We consider the (l,l)-tensor L constructed by projectively equivalent metrics g and g by (2). As it was observed in [46], see also [3, Theorem 2], it satisfies, for a certain 1-form Xi, the following equation:
V k Lij = Xigjk + Xj gik. (7)
Here and later we use g for the covariant differentiations and for the tensor manipulations with indexes. By contracting (7) with g^, we see that the 1-form Xi is the differential of the function A := 2 trace(L) = 2Lss.
Remark 1. The projectively-invariant form of this equation is due to [19], see also the survey [441 (and [12] for its two-dimensional version). It played essential role in many recent developments in the theory of projectively equivalent metrics including the solutions of two problems explicitly stated by Sophus Lie [12, 40], the proof of the discrete version of the projective Lichnerowciz conjecture [43, 51] and the proof of the Lichnerowicz conjecture for metrics of Lorenzian signature [8].
The 1-form Xi is closely related to the difference between the Levi-Civita connections of V = (r;.k) and V = (r;.k) (see e.g. [46] or [22, §2.2]): for the 1-form
& := —LiXs (8)
we have
rjk — r}k = h + 6} 0k. (9)
From formulas (8,9) we see that if A = trace(L) has zero of order k at a point p € M, then at this point the connections coincide up to the order k — 1. In particular, to any tensor field T the (k — 1)st, and also lower order, covariant derivatives of T in V and V coincide in p:
V ! V 2 ...V fc_! T =P V n V ...V T.
Let us recall one more important property of projectively equivalent metrics:
metrics and L is as in (2). Then, the Ricci curvature tensor Rij of g commutes with L, in the sense
BisLsj — LisRsj = 0. (10)
(For each x £ M the formula (10) is just the formula of the commutators of two endomorphisms of TXM: the first is given by the Ricci tensor with one index raised, and the other it given bv L).
2.3. Perturbing the metrics in the class of projectively equivalent metrics.
Let us now show that (for any k) one can perturb the metrics g and g in the class of projectively equivalent metrics such that at a point they remain the same up to order k and at another point the function A is constat up to order k.
We say that two tensors or afline connections coincide at a point p up to order k, if their
k
p
choice of a coordinate system.
p k
k
Theorem 5. Let g and g he projectively equivalent metrics and L is as in (2). Then, for each k £ N and for almost any point p £ M there exists an arbitrary small neighborhood U containing p, a point q £ U and a pair of projectively equivalent metrics g' and g' on U (whose tensor (2) will be denoted by V and the function 1 trace(L') will be denoted by A') such that the following holds:
(A) At, the point p, g coincides with g' and g coincides with g' up to order k.
A k
"Almost every point" means that the set of such points contains an open everywhere dense subset.
algebraically generic in the sense of [10, Def. 2.7]: that is, there exists a neighborhood U 3 p such that at every point of the neighborhood the number of different eigenvalues of L and the number and
the sizes of the Jordan blocks are the same (of course the eigenvalues are not necessary constant and
p
Take such a point. Note that A is the half of the sum of eigenvalues of L, counted with algebraic multiplicities. We need to find projectively equivalent metrics g' and g' such that they coincide to order k at p with g and g and such that all eigenvalues of V are constant up to order k in some
By the Splitting-Gluing construction [5, §§1.1, 1.2], it is sufficient to do this under the assumption that L has one eigenvalue, or one pair of complex-conjugated eigenvalues. If the geometric multiplicity of an eigenvalue is greater than one, by [6, Proposition 1], the eigenvalue is already a constant, so we are done since g' = g and g = g' are already as we want.
Let us now consider the case when L has one real eigenvalue of geometric multiplicity 1, or a pair of nonreal complex-conjugate eigenvalues of geometric multiplicity 1. In this case, the local structure of g and L near the point p are described in some coordinate system. There are 4 possible cases, the description was done in different papers, let us give the precise references where it can be found.
If eigenvalue is real and its geometric multiplicity is one (so the "splitted out" manifold is one-dimensional), then the description is trivial and was discussed e.g. in [5, Example in §2.1] or [39, Example 3 in §3.2.1].
If L has a pair of nonreal complex-conjugate eigenvalues of geometric multiplicity 1, then the description was done in [4, Theorem 2], see also [40, Theorem A].
If L, at each point of U, is conjugate to a Jordan block with real eigenvalue, the description is in [6, Theorem 4].
If L, at each point of U, is conjugate to a pair of Jordan blocks with complex-conjugated eigenvalues, the description is done in [6, Theorem 5].
In each of the above references, one sees that description is given by a formula and the only object we can choose is the eigenvalue(s) of L: in the 'real' case, it is a function of one variable; this function can be chosen arbitrary (with exception that one may not make it zero; though also this is allowed if we discuss not projectively equivalent metrics but 'compatible' in the terminology of [6], pairs (g,L)).
In the 'nonreal' case, the eigenvalue is a holomorphic function of one variable, again it can be chosen arbitrary (again with exception that it is never zero) in the class of holomphic functions.
In order to prove Theorem 5, one modifies the eigenvalue such that at p is coincides with the initial eigenvalue up to order k, and is constant up to order k in some other point q. One can clearly do it for any function of one variable and for any holomorphic function of one complex variable.
2.4. Carter's condition.
We will need the following result:
Theorem 6. Assume Kij is a Killing tensor for g and Rij is the Ricci curvature tensor. Suppose, at the point p € M, we have that up to order k
Vi (RisKsj — KzsRsj) = 0. (11)
Then, the Beltrami-Lapalce operator Ag and the operator K commute at the point p up to order k, that is, for every function f we have
(^AgK — KAg j f = 0 at p up to order k
Theorem above is essentially due to B. Carter. Indeed, from [15, Equation (6.16)] it follows that if Vi (^RlK? — K\Rj^ is zero at all points, then Ag and K commute at all points. Careful analysis of the arguments shows that the proof of Carter is valid also pointwise. Note that only a sketch of the proof is given in [15], and we recommend [18, §III(A)] of C. Duval and G. Valent, from which a more detailed proof can be extracted. More precisely, combining [18, Equations (3.11) and (3.16)] we obtain the above mentioned result of Carter.
2.5. If a Killing tensor vanishes up to a sufficiently high order at one point, then it is identically zero
Th eorem 7. Let M be a connected m,anifold and g be a metric of any signature on it. Assume K is a Killing tensor of order k (i.e., K is a symmetrie (0,k) tensor satisfying the equation V(iKil .ik) = 0). If K vanishes up to order k at one point, then it vanishes identically on the whole manifold.
This theorem follows from [48] (see also [25, §3]). We will need this theorem for first and second degree Killing tensors. Note that for the first degree Killing tensors (= Killing vectors, after raising the index), Theorem 7 can be obtained by the following geometric argument: if a Killing vector field vanishes at a point q up to order 1, then the flow of this vector field acts trivially on the tangent space to q. Since it commutes with the exponential mapping, the Killing vector field must
be identically zero. For second degree Killing tensors, the proof is based on the prolongation of the Killing equation which was essentially done in [50]. For all degree Killing tensors, the prolongation of Killing equation was essentially done in [48], though formally this paper discusses special case of constant curvature metrics. Indeed, for our goal the higher order terms of the prolongation are sufficient, and they do not depend on the curvature of the metric, see e.g. the discussion in [25, §3]).
3. Proof of Theorem 1.
We assume that g and g are projectively equivalent metrics of any signature on Mn, n ^ 2. We consider L given by (2), the family K(i) of Killing tensors given by (4) and the corresponding differential operators K(i). Combining Theorems 4 and 6, we see that the operators commute with Ag.
, £ R
Q :=K(t)K(s) —K(s)K(t).
Our goal is to show that it vanishes; we will first show that it is (linear) differential operator of order at most 2, i.e., that when we apply Q to a function f the higher derivatives of f vanish. This step is well-known, see e.g. [15] or [18], let us shortly recall the arguments.
Clearly, Q is a differential operator of order at most 4, since both Kand Khave order 2. One immediately sees though, that the operators K(t)KKK(i) have the same symbols, so the 4th order terms cancel when we subtract one from the other. Thus, the order of Q is at most 3. The third order terms vanish because the integrals corresponding to K( and K(i) commute by Theorem 3. Indeed, direct calculations show that the symbol of the commutator of two differential operators is the Poisson bracket of their symbols.
The proof that the first and the second order terms vanish is based on another (new) argument which will use all the results recalled in §2.
First observe that there exist a symmetric (2,0) tensor Qy and the vector field V1 such that
Q = ViQij V + ViVi.
Indeed, the operator Q does not have terms of zero order, since neither K(i) nor K « have such. One can collect all second order terms in ViQy V j and declare the rest as VlVi.
Since Ag commutes with KK, it commutes with Q. Then, Qij is a Killing (0,2) tensor
It is sufficient to show, that Qij vanishes at almost every point. It is sufficient to show this for almost every t and s. We take s and t such that the tensors K(t\ Kare nondegenerate at some point. We will work in a small neightborhood of this point, in each point of which the tensors K(t\Kare nondegenerate. Now we use Theorem 5: we first take a sufficiently big k and then, for almost every point of p of this neighborhood consider the projectively equivalent metrics g' and g' satisfying conditions (A,B) from Theorem 5.
At the point p, the metrics g and g coincide with the metrics g' and g', which implies that the Killing tensor Q'i - (i.e., the analog of the Killing tensor Qij constructed by g' and g') coincides with Qij in p. Let us show that, if k is high enough, at the point q the Killing tensor Q^ vanishes up to order 2.
At the point q, the 1-form Ai and therefore the 1-form <pi (recalled in §2.2) vanishes up to (sufficiently high) order k. Then, at the point q, the difference between Levi-Civita connections V' of </and of g' vanishes up to order k — 1, see (9). Since the Killing tensors K'(s\ K'(i) are constructed by g', g' using algebraic formulas, the covariant derivative in V' of K'(s\ K'vanishes at the point
q up to order k — 1. Then, up to the order k — 1, at the point q, the Levi-Civita connection of the (contravariant) metrics1 coincide with V'.
Then, at the point g, the Betrami-Laplace operators of the the metrics K'(s\ K'(i) coincide with K'(s\ K'(i) up to order k — 2. From the other side the Ricci tensor corresponding to the metric K'(s) commutes (in the sense
of (10)) with K'(t), up to the terms of order k — 3, since it coincides up to the terms of order k — 3 with the Ricci tensor of g' and it commutes with V and therefore with S'(t). Then, the Carter condition (11) is fulfilled up to order k — 4. Then, the operators K(i) and K(s) commute at q up to order k — 4, which means th at at q we have Q^ = 0 up to or der k — 5. If k > 7, then this implies by Theorem 5 that Q'^ is identically zero, which means it vanishes at p, where it coincides with Qij. Fin ally, Qij = 0 at p and since p was almost every point Qij = 0 on the whole manifold.
Remark 2. In fact the reader does not need to follow the precise calculations of the necessary order above: it is clear that if k is high enough then at the point q the Levi-Civita connection of the contravariant metric corresponding to
K' M (with upper indexes) coincides with that of g up to a sufficiently high order and K'(i) is parallel with respect to any of this connections u,p to a high order which means that the operators K(i) and K(s) commute at p up to some high order and Q' is zero up to a high order and is therefore identically zero.
But then Q = V1Vi, since it commutes with Ag, V1 is a Killing vector field. Using the same arguments, one shows that (for a perturbed metrics g', g'), V' 1 = 0, which implies that V1 = 0 at p. Since this is fulfilled for almost all points p, we obtain V1 = 0. Theorem 1 is proved.
4. Open problems
4.1. Introducing potential
We assume that g and g are projectively equivalent metrics of any signature on Mn. We consider the Killing tensors K « and the corresponding integrals It from Theorem 3 and ask the following questions:
Can one add functions U w : M ^ R to the integrals It such that the results still Poisson-commute? Do the corresponding differential operators, i.e., K(i) + U(t\ still commute?
Of course it is interesting to get not one example of such functions (the trivial example U(i) = const always exists) but construct all such examples, at least locally.
If L is semi-simple at almost every point (which is always the case if g is Riemannian), the answer is positive, which follows from the combination of results of [24, 17], see also [16].
4.2. Generalize the result for c-projectively equivalent metrics.
Theory of projectively equivalent metrics has a natural analogue on Kahler manifolds: theory of c-projectively equivalent metrics. Let us recall the basic definition:
Let (M,g, J) be a Kahler manifold of arbitrary signature of real dimension 2n ^ 4. A regular curve 7 : R 5 I ^ M is called if there exist functions a, ft : I ^ R such that
V.mAi(t) = ocy(t) + ft J (i(i)) for all t € I, (12)
where 7 = ^ 7.
1As explicitly indicated, we view now the Killing tensors as metrics: we first raise the indexes in (4) by g'. The result is a nondegenerate symmetric (2, 0) tensor, we view it as a contravariant metric. In order to obtain an usual
metric, with lower indexes, one needs to invert the matrix of (K.
From the definition we see immediately that the property of J-planaritv is independent of the parameterization of the curve, and that geodesies are J-planar curves. We also see that J-planar curves form a much bigger family than the family of geodesies; at every point and in every direction
J
J J J
condition that the metrics are Káhler with respect to the same complex structure is not essential; it is an easy exercise to show that if any J-planar curve of a Káhler structure (g, J) is a J-planar curve of another Káhler structure (g, J), then J = ±J.
C-projective equivalence was introduced (under the name "h-projective equivalence"or "holomor-phicallv projective correspondence") by T. Otsuki and Y. Tashiro in [45, 47]. Their motivation was to generalize the notion of projective equivalence to the Káhler situation. Otsuki and Tashiro, see also [21, §6.2], have shown that projective equivalence is not interesting in the Káhler situation, since only simple examples are possible, and suggested c-projective equivalence as an interesting object of study instead. This suggestion appeared to be very fruitful and between the 1960s and the 1970s, the theory of c-projectivelv equivalent metrics and c-projective transformations was one of the main research topics in Japanese and Soviet (mostly Odessa and Kazan) differential geometry schools. Geometric structures that are equivalent to the existence of a c-projective equivalent metric were suggested independently in different branches of mathematics, see e.g. the introductions of [42] for a list and [14] for more detailed explanation on the relation to Hamiltonian 2-forms.
It appears that many ideas and many results in the theory of projectively equivalent metrics have their counterparts in the c-projective setting. For example, the use of integrable systems in the proof of the Yano-Obata conjecture [41] about c-projective transformations is very similar to that of in the Lichnerowicz conjecture [37] for projective transformations. Compare also [7, 40]. See e.g. [8, §1.2.] for one of the explanations. In particular, Theorems 2 and 3 have clear analogs: by a c-projectivelv equivalent metric g one can construct second degree Killing tensors for g, and the corresponding integrals commute: see e.g. [13, Proposition 5.14], the result was initially obtained in [49, Theorem 2]. We ask the following question: can one generalize the result of the present paper to c-projectivelv equivalent metrics?
Do the differential operators corresponding to the Killing tensors from [13, Proposition S.l^L [4-9, Theorem 2] commute?
Also in the c-projective case, the Ricci tensor commutes with the analog of the tensor L. One can do it by the following tensor calculations which are similar to that of the proof of Theorem 4: take [20, Equation (7)] (which is the c-projective analog of [23, Equation (11)]), perturb the indexes by the trivial permutation and by the permutations ikl ^ k£i and ik£ ^ £ik and sum the results. We obtain [23, Equation (13)] (where aij corresponds to Lij in our notation). Contracting the obtained equation with g^k, we obtain an analog of (10), which implies by Theorem 6 that the operators commute with the Beltrami-Laplace operator. Unfortunately, the rest of the proof can not be directly generalized to the c-projective case, since the analog of the function A can not be a constant up to high order by [20, Corollary 3]. One can try to employ [18, Equation (3.11)] for it, but we did not manage to overcome the technical difficulties.
We do not have clear expectation how the answer would look: we tip that the operators do commute, but will not be suprised if their commutators are first order differential operators corresponding to Killing vector fields. We would like to recall here that a c-projectivelv equivalent metric allows one to construct Killing vector fields, see e.g. [8, §2] and [13, §5.2].
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12. R. Bryant, G. Manno, V. S. Matveev, A solution of S. Lie Problem,: Normal forms of 2-dim m,et,rics admitting two projective vector fields, Math. Ann. 340, 2008, no. 2, 437-463.
13. D. Calderbank, M. Eastwood, V. S. Matveev, K. Neusser, C-projective geometry, accepted to Mem. AMS, preprint arXiv: 1512.04516.
14. D. Calderbank, V. S. Matveev, S. Rosemann, Curvature and the c-projective mobility of Kaehler m,et,rics with hamiltonian 2-forms, Compositio Math. 152, 2016, 1555-1575.
15. B. Carter, Killing tensor quantum numbers and conserved currents in curved space, Phvs. Rev. D (3) 16, 1977, no. 12, 3395-3414.
16. C. Chanu, L. Degiovanni, G. Rastelli, Modified Laplace-Beltrami quantization of natural Hamiltonian system,s with quadratic constants of motion J. Math. Phvs. 58, 2017, no. 3, 033509.
17. Th. Daude, N. Kamran, F. Nicolea, Separability and Symmetry Operators for Painleve Metrics and their Conformal Deformations, SIGMA 15 (2019), 069, 42 pages. arXiv: 1903.10573.
18. C. Duval, G. Valent, Quantum integrability of quadratic Killing tensors, J. Math. Phvs. 46, 2005, no. 5, 053516, 22 pp.
19. M. Eastwood, V. S. Matveev, Metric connections in projective differential geometry, Symmetries and Overdetermined Systems of Partial Differential Equations (Minneapolis, MN, 2006, 339-351, IMA Vol. Math. Appl., 144, 2007, Springer, New York.
20. A. Fedorova, V. Kiosak, V. Matveev, S. Rosemann, The only closed Kdhler manifold with degree of mobility > 2 is (CP(n)gFvhim-study), Proc. London Math. Soc. 105, 2012, no. 1, 153-188.
21. A. R. Gover, V. S. Matveev, Projectively related metrics, Weyl nullity and metric projectively invariant equations, Proc. Lond. Math. Soc. (3) 114, 2017, no. 2, 242-292.
22. V. Kiosak, V. S. Matveev, Complete Einstein m,et,rics are geodesically rigid, Comm. Math. Phvs. 289, 2009, no. 1, 383-400.
23. V. Kiosak, V. S. Matveev, Proof Of The Projective Lichnerowicz Conjecture For Pseudo-Rie-mannian Metrics With Degree Of Mobility Greater Than Two, Comm. Mat. Phvs. 297, 2010, no. 2, 401-426.
24. B. Kruglikov, V. S. Matveev, On vanishing of topological entropy for certain integrable systems, Electron. Res. Announc. Amer. Math. Soc. 12, 2006, 19-28
25. B. Kruglikov, V. S. Matveev, The, geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta, Nonlinearitv 29, 2016, 1755-1768
26. T. Levi-Civita, Sulle trasformazioni delle equazioni dinamiche, Ann. di Mat., serie 2", 24, 1896, 255-300.
27. V. S. Matveev, P. Topalov, Trajectory equivalence and corresponding integrals, Regular and Chaotic Dynamics, 3, 1998, 30-45.
28. V. S. Matveev, P. Topalov, Metric with ergodic geodesic flow is completely determined by unparameterized geodesies, Electron. Res. Announc. Amer. Math. Soc. 6, 2000, 98-104.
29. V. S. Matveev, P. Topalov, Integrability in the theory of geodesically equivalent metrics, Kowalevski Workshop on Mathematical Methods of Regular Dynamics (Leeds, 2000). J. Phvs. A 34, 2001, no. 11, 2415-2433.
30. V. S. Matveev, Commuting operators and separation of variables for Laplacians of projectively equivalent metrics, Let. Math. Phvs., 54, 2000, 193-201.
31. V. S. Matveev, P. Topalov, Quantum integrability for the Beltrami-Laplace operator as geodesic equivalence, Math. Z. 238, 2001, no. 4, 833-866.
32. V. S. Matveev, Three-dimensional manifolds having m,et,rics with the same geodesies, Topology 42, 2003, no. 6, 1371-1395.
33. V. S. Matveev, Beltrami problem, Lichnerowicz-Obata conjecture and applications of integrable system,s in differential geometry, Tr. Semin. Vektorn. Tenzorn. Anal, 26, 2005, 214-238.
34. V. S. Matveev, Hyperbolic manifolds are geodesically rigid, Invent, math. 151, 2003, 579-609.
35. V. S. Matveev, P. Topalov, Geodesic equivalence via integrability, Geometriae Dedicata 96, 2003, 91-115.
36. В. С. Матвеев, Собственные значения отображения Синюкова для геодезически эквивалентных метрик глобально упорядочены, Матем. заметки, 77 , 2005, no.3, pp. 412-423
37. V. S. Matveev, Proof of the projective Lichnerowicz-Obata conjecture, J. Differential Geom. 75, 2007, no. 3, 459-502.
38. V. S. Matveev, P. Mounoud, Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications, Ann. Global Anal. Geom. 38, 2010, no. 3, 259-271.
39. V. S. Matveev, Geodesically equivalent metrics in general relativity, J. Geom. Phvs. 62, 2012, 675-691.
40. V. S. Matveev, Two-dimensional metrics admitting precisely one projective vector field. This paper has an Appendix Dini theorem for pseudoriemannian metrics (joint with A. Bolsinov and G. Pucacco), Math. Ann. 352, 2012, no. 4, 865-909.
41. V. S. Matveev, S. Rosemann, Proof of the Yano-Obata Conjecture for holomorph-projective transformations, J. Diff. Geom. 92, 2012, 221-261.
42. V. S. Matveev, S. Rosemann, Certification construction for Kahler manifolds and its application in c-projective geometry, Adv. Math. 274, 2015, 1-38.
43. В. С. Матвеев, О числе нетривиальных проективных преобразований замкнутых многообразий, Фундамент, и прикл. матем., 20, 2015, по.2, 125-131.
44. V. S. Matveev, Projectively invariant objects and the index of the group of affine transformations in the group of projective transformations, Bull. Iran. Math. Soc. 44, 2018, 341-375.
45. T. Otsuki, Y. Tashiro, On curves in Kaehlerian spaces, Math. J. Okavama Univ. 4, 1954, 57-78.
46. H. С. Синюков, Геодезические отображения римановых пространств, "Наука", Москва, 1979, MR0552022, Zbl 0637.53020.
47. Y. Tashiro, On a holomorphically projective correspondence in an almost complex space, Math. J. Okavama Univ. 6, 1957, 147-152.
48. G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phvs. 27, 1986, no. 11, 2693-2699.
49. P. Topalov, Geodesic hierarchies and involutivity, J. Math. Phvs. 42, 2001, no. 8, 3898-3914.
50. Th. Wolf, Structural equations for Killing tensors of arbitrary rank, Comput. Phvs. Comm. 115, 1998, 316-329.
51. A. Zeghib, On discrete projective transformation groups of Riemannian manifolds, Adv. Math. 297, 2016, 26-53.
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32. Matveev, V. S. 2003, "Three-dimensional manifolds having metrics with the same geodesies", Topology vol. 42, no. 6, pp. 1371-1395.
33. Matveev, V. S. 2005, "Beltrami problem, Lichnerowicz-Obata conjecture and applications of integrable systems in differential geometry", Tr. Semin. Vektorn. Tenzorn. Anal, vol. 26, pp. 214-238.
34. Matveev, V. S. 2003, "Hyperbolic manifolds are geodesically rigid", Invent, math., vol. 151, pp. 579-609.
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38. Matveev, V. S. к Mounoud, P. 2010, "Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications", Ann. Global Anal. Geom., vol. 38, no. 3,pp. 259-271.
39. Matveev, V. S. 2012, "Geodesically equivalent metrics in general relativity", J. Geom. Phys., vol. 62, pp. 675-691.
40. Matveev, V. S. 2012, "Two-dimensional metrics admitting precisely one projective vector field. This paper has an Appendix Dini theorem for pseudoriemannian metrics (joint with A. Bolsinov and G. Pucacco)", Math. Ann., vol. 352, no. 4, pp. 865-909.
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49. Topalov, P. 2001, "Geodesic hierarchies and involutivitv", J. Math. Phys., vol. 42, no. 8, pp. 3898-3914.
50. Wolf, Th. 1998, "Structural equations for Killing tensors of arbitrary rank", Comput. Phys. Comm., vol. 115, pp. 316-329.
51. Zeghib, A. 2016, "On discrete projective transformation groups of Riemannian manifolds", Adv. Math., vol. 297, pp. 26-53.
Получено 11.12.2019 г.
Принято в печать 11.03.2020 г.
