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ЧИСЛА ТАЧИБАНЫ ЗАМКНУТЫХ МНОГООБРАЗИЙ С ЗАЩЕМЛЕННОЙ ОТРИЦАТЕЛЬНОЙ СЕКЦИОННОЙ КРИВИЗНОЙ

Научный труд разместил:
Baizag
11 сентября 2020
Автор: Степанов С. Е.

УДК 514.764

S. Е. Stepanov] , 1.1. Tsyganok2

1 2 Financial University under the Government of the Russian Federation s. e.stepanov@mail.ru doi: 10.5922/0321-4796-2020-51-13

On the Tachibana numbers of closed manifolds with pinched negative sectional curvature

Conformal Killing form is a natural generalization of conformal Killing vector field. These forms were extensively studied by many geometricians. These considerations were motivated by existence of various applications for these forms. The vector space of conformal Killing ¿»-forms on an n-dimensional (l < p < n -1) closed Riemannian manifold M has a finite dimension tp (m ) named the Tachibana number. These numbers are conformal scalar invariant of M and satisfy the duality theorem: tp (m) = tn- (m ).

In the present article we prove two vanishing theorems. According to the first theorem, there are no nonzero Tachibana numbers on an n-dimensional (n > 4) closed Rie-mannian manifold with pinched negative sectional curvature such that -1 - ^ < sec < -1 for some pinching con«,,&<(" -1)". Fom the second theorem we conclude

that there are no nonzero Tachibana numbers on a three-dimensional closed Riemannian manifold with negative sectional curvature.

Поступила в редакцию 22.02.2020 г. © Stepanov S. E., Tsyganok I. I., 2020

1. Introduction and results

Conformal Killing p-forms or, in other words, conformal Killing — Yano p-tensors have been defined on n-dimensional Rie-mannian manifolds (1 < p < n -1) more than fifty years ago by S. Tachibana and T. Kashiwada (see [1; 2]) as a natural generalization of conformal Killing vector fields. Since then these forms were extensively studied by many geometricians. These considerations were motivated by existence of various applications for these forms (see, for example, [3; 4]).

The vector space of conformal Killing p-forms on an n-dimensional closed (i.e. compact without boundary) Riemannian manifold (M, g) has a finite dimension tp (M) named the Tachibana

number (see [5—7] and etc.). These numbers t1 (M), ..., tn-1 (M) are conformal scalar invariant of (M, g) and satisfy the duality theorem: tp (M ) = tn- (M). The theorem is an analog of the well

known Poincare duality theorem for the Betti numbers of closed (M, g). Moreover, we proved in [7] that Tachibana numbers t1 (M), ...,tn-1 (M) are equal to zero for an n-dimensional (n >2) closed Riemannian manifold (M, g) with negative curvature operator R : SqM ^ SlM defined on the vector bundle of traceless symmetric tensor fields of order 2 (see [8]). Based on this theorem, we prove here the following theorem.

Theorem 1. Let (M, g ) be an n-dimensional (n > 4) closed Riemannian manifold with pinched negative sectional curvature. Suppose that -1< sec < -1 for an arbitrary ^ <(n l) ; then Tachibana numbers t1 (M), ..., tn-1 (M) of (M, g) are equal to zero.

Remark. We recall here that for every n > 4 there exists a closed n-dimensional Riemannian manifold (M, g) such that its

sectional curvatures satisfy the inequalities - H < sec <-1 for some pinching constant H, but M does not admit a metric g of constant negative sectional curvature (see [9]).

For the case when n = 3 the following theorem is true.

Theorem 2. If (M, g) is an 3-dimensional closed Riemannian manifold with negative sectional curvature then its Tachibana numbers t1 (M) and t2 (M) are equal to zero.

2. Proofs of Theorems

Let (M, g) be an n -dimensional (n > 2) Riemannian manifold and S0M is a vector bundle of symmetric traceless 2-forms. For any point x eM there exists an orthonormal eigen-frame e1,..., en of TxM such that p- = px (ei, e- )= /ui Sij for any pe C ) and for the Kronecker delta SiJ. Then we have the formula (see [10, p. 388])

R pkpk + R-u plpk = 2Xsec(ei ae-)(U -Uj)2 (21)

i < j

where sec (ei ae-)= R (et,e-,et,e-) is the sectional curvature sec <jx of (M, g) in the direction of the tangent two-plane section ax = span \\et,e-} at x eM. In turn, the components of the curvature tensor R and the Ricci tensor Ric are denoted by R-kl = R (e,,ej,ek,el) and Rij = Ric(ei,ej), respectively.

If in addition, suppose that there is a point x e M where all sectional curvatures satisfies the double inequality - A < sec < -B for some constant A > B > 0, then from equation (2.1) we obtain the inequality (see [11])

- nApJPij < R1jpkpk + Rjkiplpj <-nBplPi- (2 2)

for an arbitrary nonzero px e S0p (T*M). In addition, we have the double inequality

-(n - 1)ApJpj < R p&k\\k <-(n - i)B\\3p>1] (2.3)

If we suppose that ||p||2 = p1pJj, then from (2.2) and (2.3) we obtain the following inequalities

(- nA + (n -1) B )\\\\p\\\\2 < g( R (p),p) <(- nB + (n -1) A)\\\\p\\\\2, where g( R(p), p) = Rjki p&&pk for the curvature operator

r : s2 (T:M So2 (T:M ).

Let A = sB, then we have g(R(p),p) < 0 for any nonzero

pe C"(so2M) for an arbitrary 0<s< 1 + (n-1) 1. Theorem 1 is proved.

In dimension three we have (see [12])

Ryki = gikRji- giiRjk- gjkRii + gjiRik- Xs (gikgji- giigjk).

On the other hand, we can always diagonalize the Ricci tensor Ric at an arbitrary point x e M, so that R = A1S1J- where Al, X2, X3

are its eigenvalue. Then in three dimensions the only nonzero components of the curvature tensor R are components of the form

R1212 = % ( +^2 ); R1313 = X (A +^3 ) ;

R2323 = X (A +^3 ) .

Thus, the condition for negative sectional curvature in three dimensions is that each eigenvalue of the Ricci tensor is bigger than sum of other two. Then we have the proposition (compare this statement with Corollary 8.2 of [12]).

Proposition. In dimension three a metric g has negative sec-tionai curvature if and oniy if Ric > Y2 s g for the scaiar curvature s = tracegRic.

Then in dimension three we can deduce the inequalities

g(R(p),p) < 12si PI2 < 0

for a nonzero pe C""(s^M). In this case, the curvature operator

R : S02M ^ S02M is negative definite. At the same time, the vanishing theorem from the paper [7] is the stating that Tachibana numbers t1 (M) = 0, ..., tn-1 (M) = 0 for an n-dimensional (n >) closed Riemannian manifold (M, g) with negative curvature operator

R : S02M S02M . Then in dimension three we have t1 (M ) = = t2 (M) = 0. Thus, Theorem 2 is proved.

References

1. Kashiwada, T.: On conformal Killing tensor. Natural. Sci. Rep. Ochanomizu Univ., 19:2, 67—74 (1968).
2. Tachibana, S.: On conformal Killing tensor in a Riemannian space. Tohoku Math. J., 21, 56—64 (1969).
3. Benn, M., Charlton, P.: Dirac symmetry operators from conformal Killing — Yano tensors. Class. Quantum Grav., 14, 1037—1042 (1997).
4. Stepanov, S. E.: On conformal Killing 2-form of the electromagnetic field. J. of Geom. and Phys., 33:3-4, 191—209 (2000).
5. Stepanov, S. E., Mikes, J.: Betti and Tachibana numbers of compact Riemannian manifolds. Diff. Geom. and its Appl., 31:4, 486—495 (2013).
6. Stepanov, S.E.: Curvature and Tachibana numbers. Sb. Math., 202:7, 135—146 (2011).
7. Stepanov, S.E., Tsyganok, I. I.: Theorems of existence and of vanishing of conformally Killing forms. Russian Mathematics, 58:10, 46—51 (2014).
8. Bourguignon, J. P., Karcher, H.: Curvature operators: pinching estimates and geometric examples. Ann. Sc. Ec. Norm. Sup. Paris, 11, 71—92 (1978).
9. Gromov, M., Thurston, W.: Pinching constants for hyperbolic manifolds. Invent. Math., 89, 1—12 (1987).
10. Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. J. Diff. Geom., 3, 379—392 (1969).
11. Rovenski, V., Stepanov, S.E., Tsyganok, I. I.: On the Betti and Tachibana numbers of compact Einstein manifolds. Mathematics, 7:12, 1210 (2019).
12. Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Diff. Geom., 17, 255—306 (1982).

С. E. Степанов1 , И. И. Цыганок2 12 Финансовый университет при Правительстве РФ, Россия s. e.stepanov@mail.ru doi: 10.5922/0321-4796-2020-51-13

Числа Тачибаны замкнутых многообразий с защемленной отрицательной секционной кривизной

Поступила в редакцию 22.02.2020 г.

Конформная форма Киллинга является естественным обобщением конформного векторного поля Киллинга. Эти формы широко изучались многими геометрами, что было мотивировано существованием различных приложений для этих форм. Векторное пространство конформных /»-форм Киллинга имеет на замкнутом «-мерном (1 < р < п -1) римановом многообразии М конечную размерность Iр (м), называемую числом Тачибана. Эти числа являются конформными скалярными инвариантами многообразия и удовлетворяют теореме двойственности tр (м) = tп- (м).

В данной статье мы доказываем две «теоремы исчезновения». В соответствии в первой теоремой не существует ненулевых чисел Тачибаны на п-мерном (п > 4) замкнутом римановом многообразии с защемленной отрицательной секционной кривизной такой, что

-1 - 8 < 5ес < -1 ддя постоянной 8 <(п 1) . Согласно второй теореме не существует ненулевых чисел Тачибаны на трехмерном замкнутом римановом многообразии с отрицательной секционной кривизной.

Список литературы

1. Kashiwada T. On conformai Killing tensor // Natural. Sci. Rep. Ochanomizu Univ. 1968. Vol. 19, №2. Р. 67—74.
2. Tachibana S. On conformal Killing tensor in a Riemannian space // Tohoku Math. J. 1969. №21. Р. 56—64.
3. Benn M., Charlton P. Dirac symmetry operators from conformal Killing — Yano tensors // Class. Quantum Grav. 1997. № 14. Р. 1037— 1042.
4. Stepanov S. E. On conformal Killing 2-form of the electromagnetic field // J. of Geom. and Phys. 2000. Vol. 33, № 3-4. P. 191—209.
5. Stepanov S. E., Mikes J. Betti and Tachibana numbers of compact Riemannian manifolds // Diff. Geom. and its Appl. 2013. Vol. 31, № 4. P. 486—495.
6. Stepanov S. E. Curvature and Tachibana numbers // Mat. Sb. 2011. Vol. 202, № 7. P. 135—146.
7. Stepanov S. E., TsyganokI.I. Theorems of existence and of vanishing of conformally killing forms // Russian Mathematics. 2014. Vol. 58, № 10. Р. 46—51.
8. Bourguignon J.P., Karcher H. Curvature operators: pinching estimates and geometric examples // Ann. Sc. Ec. Norm. Sup. Paris. 1978. № 11. Р. 79—92.
9. Gromov M., Thurston W. Pinching constants for hyperbolic manifolds // Invent. Math. 1987. № 89. Р. 1—12.
10. Berger M., Ebin D. Some decompositions of the space of symmetric tensors on a Riemannian manifold // J. Diff. Geom. 1969. № 3. Р. 379—392.
11. Rovenski V., Stepanov S.E., Tsyganok I.I. On the Betti and Tachibana numbers of compact Einstein manifolds // Mathematics. 2019. Vol. 7, № 12. Р. 1210.
12. Hamilton R. S. Three-manifolds with positive Ricci curvature // J. Diff. Geom. 1982. № 17. Р. 255—306.
РИМАНОВО МНОГООБРАЗИЕ КОНФОРМНЫЙ ТЕНЗОР КИЛЛИНГА ЯНО СЕКЦИОННАЯ КРИВИЗНА ТЕОРЕМА ИСЧЕЗНОВЕНИЯ riemannian manifold conformal killing yano tensor sectional curvature vanishing theorem