Decomposition theorems of conformal Killing forms on totally umbilical submanifolds Текст научной статьи по специальности «Математика»

Список литературы

1. Малаховский В. С. Теория конгруэнций кривых и поверхностей второго порядка в трехмерном проективном пространстве. Калининград, 1986.

2. Фиников С. П. Теория конгруэнций. М. ; Л., 1950.

V. Malakhovsky, E. Yurova

Congruences of quadrics in three-dimensional projective space associated with pair of surfaces

Two-parametric family (congruence) K2 of quadrics Q in three-dimensional projective space P3 is investigated, possessing the following properties: on each quadric Q e K2 there are two different focal points A1 and A2 at

which focal tangents intersect at one point A0 and are the asymptotic tangents of the surface (A0), and the tangents to the curves on the surface (Aj) that corresponds the focal curves on the surface (Aj) (i, j, k = 1, 2; i Ф j ) also intersect at one point A3 and are the asymptotic tangents of the surface (A3), mareover the asymtotic curves that envelop A0At and A3Aj are correspond, and A0 and A3 are polar conjugated.

UDC 514.764

J. Mikes

Palacky University, Olomouc, Czech Republic

S.E. Stepanov, I.I. Tsyganok

Finance University under the Government of Russian Federation

Decomposition theorems of conformal Killing forms on totally umbilical submanifolds

A Riemannian manifold of positive curvature operator has been studied from many directions. It is well known, that an «-dimensional closed Riemannian manifold with positive curvature operator ^ is a spherical space form and its Betti numbers b1(M'), ..., b „_ 1(M ') are zero. In addition, we proved that on an

«-dimensional closed Riemannian manifold (M, g) with positive curvature operator ^ an arbitrary conformal Killing r-form a is uniquely decomposed in the form a' + a" where a' is a Killing r-form and a "'is a closed conformal Killing r-form on (M, g) for all r = 1, ..., n - 1. In the present paper we prove three decomposition theorems of conformal Killing forms on totally umbilical submanifolds in Riemannian manifolds.

Key words: conformal Killing form, decomposition theorem, totally umbilical submanifold, Riemannian manifolds.

1. Let R be the covariant curvature tensor of a Riemannian manifold (M, g). The relation (see: [1, p. 36])

g((( a Y), V a W) = R(X A Y,V A W) = R{X, Y, W, V)

defines a self-adjoint symmetric operator ^ : A2M ^ K'M . This operator is called the curvature operator of (M, g).

Let {e1 ,...,en} be an orthogonal basis in TxM at an arbitrary point x e TxM such that et a ej diagonalize the curvature operator <H(ei a ej )= ay (x) ei a ej. We say that a Riemannian manifold

(M, g) has positive curvature operator if all its eigenvalues are positive. Note that this definition is invariant, because it does not depend on the choice of the basis {e1 ,...,en} atx e TJM . If the operator curvature ^ is positive-definite at each point x e M then we say that ^ is positive-definite on the manifold (M, g).

Next, we say that the positive curvature operator ^ is bounded from below on (M, g) if all the eigenvalues of ^ greater than or equal to some positive number A at all points of (M, g). Side by side, we recall that if the positive curvature operator ^ of a simply connected closed manifold (M, g) is bounded from below then (M, g) is homeomorphic to a sphere (see: [2]).

The normalized quadratic form sec(jr) = g(^(Xx a Yx^ Xx a Yx)

W g(Xx a Yx, Xx a Yx)

is called the sectional curvature of two-plane

n = span{Xx,Yx}czTJM . In addition, we note (see: [1], p. 63) that

if {,...,en} is an orthogonal basis for TxM such that ei a ej di-

agonalize the curvature operator ei a e} )= Aij (x) ei a e} then for

any two-plane n in TXM we have sec(n) e [min AiJ (x), max AiJ (x).

Moreover (see: [3]), if the eigenvalues of ^ are Ai]- (o)>l(o)> 0

then the sectional curvatures are Xij(o)>A(o)l2>0 at x eM.

Therefore, if the positive curvature operator ^ of (M, g) is bounded from below then the sectional curvature of this manifold is positive and bounded from below too. On the other hand, we recall that a complete Riemannian manifold (M, g) is closed if its sectional curvature is bounded from below by some positive number (see: [4], p. 212—213).

Thus, using all these facts, we can formulate the following theorem.

Theorem 1. Let (M', g') be an n '-dimensional simply connected and complete non-totally geodesic, totally umbilical submanifold in an n-dimensional Riemannian manifold (M, g). If the curvature operator ^ of (M, g) is positive semi-definite on the bundle A2M/ over the submanifold (M', g') and the mean curvature H2 of (M', g') reaches

its lowest value H2min, then (M', g) is closed and homeomorphic to a sphere, its Betti numbers b1(M), ..., b n>_ 1(M) are zero and an arbitrary conformal Killing r-form c on (M', g') can be decomposed in the form c' + c" where c'is a Killing r-form and c" is a closed con-formal Killing r-form for all r = 1, ..., n' -1.

Proof We consider a complete and simply connected non-totally geodesic, totally umbilical submanifold (M ', g') of a Riemannian manifold (M, g). The tensor R' of (M ', g') we can find from the Gauss curvature equation, which for a totally umbilical (M , g ) in a Riemannian manifold (M, g) has the form

R' (, Y', V', W') = Rf X', fY', fV', fW')+ + H2 (g' (X ' ,W' ) g '(Y ',V')-g '(Y' ,W' ) g '(X' ,V')) (1)

for the mean curvature H2 = g (hh) of the submanifold (M , g')

and any vector fields X ' ,Y' Z',W' e CrDTM '. We can rewritten the Gauss curvature equation (1) in the following form

g '(((e'), e')=g '(<R(f "e') f "e') + 2H 2|| ef (2)

for an arbitrary ff e CXA2M ' and || e'f = g' (e',e').

Next, we suppose that there exists the positive number H2riiin which is the lowest value of the mean curvature H2 of (M", g') and g '(n(f "e'), f "e')> 0 for an arbitrary ff e C"AM' then from (2) we obtain the inequalities

g ' (^ '(q '), e' )> 2H;;iii|| e' |2 > o. (3)

In turn, from (3) we conclude that the curvature operator of (M , g ) is positive and bounded from below and hence the sectional curvature sec O of (M ; g') is positive and bounded from below too. In this case, the simply connected complete manifold (M , g ) is compact and homeomorphic to a sphere. Then its Betti numbers b1(M ), ., bn ' _ 1(M ) are zero and an arbitrary conformal Killing r-form a on (M , g') is uniquely decomposed in the form a' + a" where a' is a Killing r-form and a" is a closed conformal Killing r-form on (M , g') for all r = 1, ., n/- 1 (see: [5]).

2. From the Gauss curvature equation (1) we can obtain identities relating sectional curvatures of the Riemannian manifold (M, g) and its totally umbilical submanifold (M , g ) and the mean curvature of (M , g ) (see also: [6])

sec ' {n) = sec(^) + H2 (4)

h T M' d ( ) g(^(f*X'x a fX), f"X'x a fX)

where n c TM and sec\n) = v /v—x—-—^^—x— .

g ((".X'x a fX,f*X'x a fY)

Schouten proved (see: [7], p. 301) that every totally umbilical submanifold of dimension > 4in conformally flat Riemannian manifold is conformally flat. For these manifolds the following proposition is true.

Corollary 2. Let (M', g ) be an n '-dimensional (n' > 4) simply connected and complete non-totally geodesic, totally umbilical sub-manifold of an n-dimensional conformally flat Riemannian manifold (M, g). If the sectional curvature sec (n) of (M, g) is positive semi-define for all two-plane section n of TM/ and the mean curvature H2 of (M', g) reaches its lowest value H2min, then (M', g) is confor-

mally diffeomorphic to a sphere, its Betti number bi(M), ..., bni(M ) are zero and an arbitrary conformal Killing r-form c can be decomposed in the form c' + c" where c'is a Killing r-form and c" is a closed conformal Killing r-form on (M', g ) for all r = 1, ., n' - 1.

Proof. Now, we consider a simply connected and complete non-totally geodesic, totally umbilical conformally flat submanifold (M', g') of a Riemannian manifold (M, g). Next, we suppose that there exists the

positive number Hm2 in which is the lowest value of the mean curvature H2 of (M , g') and sec(n) > 0 for all two-plane section n of TM'then from (4) we obtain the inequalities sec ' (n) = sec (n)+h 2 > h min > 0.

In this case, the simply connected complete manifold (M , g ) is closed. In addition, we recall that a conformally flat simply connected closed Riemannian manifold (M ', g ) is conformally diffeomorphic to a sphere (see: [8]). In this case, Betti numbers b\(M ), ., bn '_ j(M ) of (M ', g ) are zero and an arbitrary conformal Killing r-form c on (M ', g') is uniquely decomposed in the form c' + c" where c' is a Killing r-form and c" is a closed conformal Killing r-form on (M', g') for all r = 1, ., n/- 1. 3. The Ricci curvature of (M ', g ) is

Ric(YX,Z') = trace (( ^ R(X',Yx')Z') = (R(,Y')Z',ei)

i=1

for an orthonormal basis {el,e2,...,en} at an arbitrary point x e M'.

n

If X X = ej, then Ric(X' ,X' ) = ^ sec (X' ,ei) .In dimension 3 we

i=2

have the relation (see: [1, р. 38]) 92

Ric ' (e1 ,e1 ) = sec ' (n12 )+sec ' (n13 ); Ric ' (e2 ,e2 ) = sec ' (n12 ) + sec ' (n23 ) ; Ric ' (e3 ,e3 ) = sec ' (n23 ) + sec ' (n13 )

where n12 = span{e1,e2}, n23 = span{e2,e3} and tt13 = span{e1 ,e3}.

This means the sectional curvature sec' (n) can be computed from Ric' and the sectional curvature determine Riemannian curvature operator M'. Moreover, if (x)>^3 (x)>^3 (x) are eigenvalues of M with respect to some orthonormal basis {e1 ,e2 ,e3} at an arbitrary point x e M' then we have (see: [1, p. 61]) ' À12 ( x ) 0 0 ^

M x = 0 A23 (x ) 0

0 0 ^(x)

Ric'x = -x 2

' ^23 ( x ) + ^3 ( x ) 0 0

0

À2 (x)+ ^3 (x)

0 ^ (x )

-^23 (x)

Based on these facts and Theorem 1, we can prove the following corollary.

Corollary 3. Let (M, g') be a three-dimensional simply connected and complete non-totally geodesic, totally umbilical sub-manifold of an n-dimensional Riemannian manifold (M, g). If the sectional curvature sec (n) of (M, g) is positive semi-define for all

two-plane section n of TM'and the mean curvature H2 of (M', g )

reaches its lowest value Hmin, then (M', g ) is closed and homeo-morphic to a sphere, its Betti numbers bi(M ) and b2(M) are zero and an arbitrary conformal Killing r-form a can be decomposed in the form a' + a" where a' is a Killing r-form and a" is a closed conformal Killing r-form on (M', g ) for r = 1, 2.

Remark. If (M , g ) is closed and b\(M) = 0 then any closed conformal Killing 1-form a has the form a = d f for a smooth

scalar function f such that Vd f = -n lA f ■ g where A isa

Laplacian — Beltrami operator on (M , g ). In this case (M , g ) is conformally diffeomorphic to a sphere (see: [10]). Therefore in every our proposition a simply connected and complete totally umbilical submanifold (M , g )is conformally diffeomorphic to a sphere if (M , g ) admits a non-Killing conformal Killing 1-form.

Acknowledgments. The paper was supported by grant P201/11/0356 of The Czech Science Foundation.

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Й. Микеш, С. Е. Степанов, И. И. Цыганок

Теормы разложения конформно килинговых форм на тотально омбилических подмногообразиях

Доказываются теоремы о разложении конформно киллинговой г-формы в ортогональную сумму киллинговой и замкнутой конформно киллинговой г-форм на вполне омбилических поверхностях «-мерного риманова многообразия (г = 1, ..п — 1).

УДК 514.76

Н. Д. Никитин, О. Г. Никитина

Пензенский государственный университет

Инфинитезимальные преобразования аффинной связности касательного расслоения пространства нелинейной связности

В работе показано, что полный лифт X инфинитезималь-ного преобразования X дифференцируемого многообразия М оставляет инвариантным аффинную связность касательного расслоения Т(М) пространства нелинейной связности тогда и только тогда, когда векторное поле X является инфинитези-мальным движением в пространстве нелинейной связности.

Ключевые слова: касательное расслоение, нелинейная связность, полный лифт векторного поля, производная Ли, инфинитезимальное аффинное движение.

Пусть М — «-мерное дифференцируемое многообразие, Т(М) — касательное расслоение, п: Т(М) ^ М — каноническая проекция, О = Я \ {0} группа Ли относительно операции умножения, действующая на касательном расслоении по закону: для любого а е О преобразование Яа : Т(М) ^ Т(М) отображает произвольный элемент г е Т(М) в Яа (г) = аг, где