147 Kragujevac J. Math. 25 (2003) 147 154. CONDITIONS FOR INVARIANT SUBMANIFOLD OF A MANIFOLD WITH THE (ϕ, ξ, η, G)-STRUCTURE Jovanka Nkć Faculty of Techncal Scences, Unversty of Nov Sad, Trg Dosteja Obradovća 6, 21000 Nov Sad, Serba and Montenegro Abstract. We shall nvestgate a necessary and suffcent condton for a submanfold mmersed n an almost paracontact Remannan manfold to be nvarant and show further propertes of nvarant submanfold n a manfold wth the (ϕ, ξ, η, G)-structure. 1. (ϕ, ξ, η, G)-structure Let M be an m-dmensonal dfferentable manfold. If there exst on M a (1, 1)-tensor feld ϕ, a vector feld ξ and a 1-form η satsfyng η(ξ) = 1, ϕ 2 = I η ξ, (1.1) where I s the dentty, then M s sad to be an almost paracontact manfold. In the almost paracontact manfold, the followng relatons hold good: ϕξ = 0, η ϕ = 0, rank (ϕ) = m 1. (1.2) Every almost paracontact manfold has a postve defnte Remannan metrc G such that η( X) = G(ξ, X), G(ϕ X, ϕȳ ) = G( X, Ȳ ) η( X)η(Ȳ ), X, Ȳ X ( M), (1.3)
148 where X ( M) denotes the set of dfferentable vector felds on M. In ths case, we say that M has an almost paracontact Remannan structure (ϕ, ξ, η, G) and M s sad to be an almost paracontact Remannan manfold [1]. From (1.3) we can easly get the relaton G(ϕ X, Ȳ ) = G( X, ϕȳ ). (1.4) Hereafter, we assume that M s an almost paracontact Remannan manfold wth a structure (ϕ, ξ, η, G). It s clear that the egenvalues of the matrx (ϕ) are 0 and ±1, where the multplcty of 0 s equal to 1. Let M be an n-dmensonal dfferentable manfold (m n = s) and suppose that M s mmersed n the almost paracontact Remannan manfold M by the mmerson : M M. We denote by B the dfferental of the mmerson. The nduced Remannan metrc g of M s gven by g(x, Y ) = G(BX, BY ), X, Y X (M), where X (M) s the set of dfferentable vector felds on M. We denote by T p (M) the tangent space of M at p M, by T p (M) the normal space of M at p and by {N 1, N 2,..., N s } an orthonormal bass of the normal space T p (M). The transform ϕbx of X T p (M) by ϕ and ϕn of N by ϕ can be respectvely wrtten n the next forms: ϕbx = BψX + u (X)N, X X (M), (1.5) ϕn = BU + λ j N j, (1.6) where ψ, u, U and λ j are respectvely a nduced (1, 1)-tensor, 1-forms, vector felds and functons on M and Latn ndces h,, j, k, l run over the range {1, 2,..., s}. The vector feld ξ can be expressed as follows: ξ = BV + α N, (1.7) where V and α are respectvely a vector feld and functons on M. Usng (1.5) and (1.6) for X, Y X (M) g(ψx, Y ) = G(BψX, BY ) = G(ϕBX, BY ) = G(BX, ϕby ) = G(BX, BψY ). Therefore we have g(ψx, Y ) = g(x, ψy ). Furthermore, from G(ϕBX, N ) = G(BX, ϕn ) and G(ϕN, N j ) = G(N, ϕn j ), we can respectvely get the equatons u (X) = g(u, X), λ j = λ j.
149 Lemma 1.1. In a submanfold M mmersed n an almost paracontact Remannan manfold M, the followng equatons hold good: ψ 2 X = X v(x)v u (X)U or ψ 2 = I v V u U, X X (M), (1.8) u j (ψx) + λ j u (X) + α j v(x) = 0, (1.9) ψu j + λ j U + α j V = 0, (1.10) u k (U j ) = δ kj α k α j λ k λ j, (1.11) where v s a 1-form on M and v(x) = g(v, X). Proof. From (1.5), we have ϕ 2 BX = ϕbψx + u (X)ϕN = Bψ 2 X + u (ψx)bu + u (X) j λ j N j. On the other hand, snce we have ϕ 2 BX = BX η(bx)ξ = BX v(x) v(x) j α j N j, we get (1.8) and (1.9). Smlarly, from (1.6) we have (1.10) and (1.11). Equatons (1.9) and (1.10) are equvalent. Lemma 1.2. In a submanfold M mmersed n an almost paracontact Remannan manfold M, the followng equatons hold good: ψv + α U = 0, u (V ) + α j λ j = 0, (1.12) j=1 v(v ) = 1 α 2, (1.13) g(ψx, ψy ) = g(x, Y ) v(x)v(y ) u (X)u (Y ), (1.14) where X, Y X (M).
150 Proof. From (1.7), ϕξ = ϕbv + α ϕn = BψV + u (V )N + α (BU + j λ j N j ). By means of ϕξ = 0, we have (1.12). Smlarly, from η(ξ) = 1 we get (1.13). And usng G(ϕBX, ϕby ) = G(BX, BY ) η(bx)η(by ), we have (1.14). Let {N 1, N 2,..., N s } be an orthonormal bass of the normal space T p (M) at p M [1]. We assume that { N 1, N 2,..., N s } s the another orthonormal bass of T p (M) and we put By means of G( N, N j ) = s N = k l N l. (1.15) k l k lj = δ j, from whch s k h k jh = δ j. Consequently a matrx (k j ) s an orthogonal matrx. Thus from (1.15), we have N j = s k jl Nl. Makng use of (1.15), equatons (1.5), (1.6) and (1.7) are respectvely wrtten n the followng form: where ϕbx = BψX + ϕ N l = BŪl + ξ = BV + ū l (X) N l, h=1 λ lh Nh, (1.16) h=1 ᾱ l Nl, ū l (X) = k l u (X), Ū l = k l U, (1.17) λ lh =,j=1 k l λ j k jh, λlh = λ hl, (1.18) ᾱ l = k l α. By vrtue of (1.17), the lnear ndependence of vectors U ( = 1, 2,..., s) s nvarant under the transformaton (1.15) of the orthonormal bass {N 1, N 2,..., N s }.
151 Furthermore, because λ j s symmetrc n and j, from (1.18) we can fnd that under a sutable transformaton (1.15) λ j reduces to λ j = λ δ j, where λ ( = 1, 2,..., s) are egenvalues of matrx (λ j ). In ths case, (1.16) and (1.11) are respectvely wrtten n the next forms: ϕ N l = BŪl + λ l Nl, ū k (Ūj) = δ kj ᾱ k ᾱ j λ k λ j δ kj, (1.19) from whch we have ū j (Ūj) = 1 ᾱ 2 j λ2 j and ū k(ūj) = ᾱ k ᾱ j (k j). 2. Invarant submanfolds of an almost paracontact Remannan manfold Let M be a submanfold mmersed n an almost paracontact Remannan manfold M. If ϕ(b(t p (M))) T p (M) for any pont p M, then M s called an nvarant submanfold [4]. In an nvarant submanfold M, (1.5), (1.6) and (1.7) are respectvely wrtten n the followng forms: ϕbx = BψX, X X (M), (2.1) ϕn = λ j N j, (2.2) j=1 ξ = BV + α N. (2.3) Furthermore, from Lemma 1.1 and Lemma 1.2, we have the followng lemmas. Lemma 2.1. In an nvarant submanfold M mmersed n an almost paracontact Remannan manfold M, the followng equatons hold good: ψ 2 = I v V, (2.4) α V = 0, (2.5) δ kj α k α j λ k λ j = 0, (2.6) ψv = 0, (2.7)
152 α λ j = 0, (2.8) v(v ) = 1 α 2, (2.9) g(ψx, ψy ) = g(x, Y ) v(x)v(y ), X, Y X (M). (2.10) From (2.5) and (2.9), we get the followng two cases: When V = 0 (or α 2 = 1), that s, ξ s normal to M, snce from (2.4) and (2.10) we have ψ 2 = I, g(ψx, ψy ) = g(x, Y ), (ψ, g) s an almost product Remannan structure whenever ψ s non-trval. When V 0 (or α = 0), that s, ξ s tangent to M, by means of (2.4), (2.9), (2.10) and v(x) = g(v, X), (ψ, V, v, g) s an almost paracontact Remannan structure. From [2], [3] we have Theorem 2.1. Let M be an nvarant submanfold mmersed n an almost paracontact Remannan manfold M wth a structure (ϕ, ξ, η, G). Then one of the followng cases occurs. 1.) ξ s normal to M. In ths case, the nduced structure (ψ, g) on M s an almost product Remannan structure whenever ψ s non-trval. 2.) ξ s tangent to M. In ths case, the nduced structure (ψ, V, v, g) s an almost paracontact Remannan structure. Furthermore, we have the followng theorems. Theorem 2.2. In order that, n an almost paracontact Remannan manfold M wth a structure (ϕ, ξ, η, G), the submanfold M of M s nvarant, t s necessary and suffcent that the nduced structure (ψ, g) on M s an almost product Remannan structure whenever ψ s non-trval or the nduced structure (ψ, V, v, g) on M s an almost paracontact Remannan structure. Proof. From Theorem 2.1, the necessty s evdent. Conversely, we frst assume that the nduced structure (ψ, g) s an almost product Remannan structure. Then from (1.8) we have v(x)v + u (X)U = 0, from whch g(v(x)v + u (X)U, X) = 0, that s, v(x) 2 + u (X) 2 = 0. Consequently, snce we get v(x) = u (X) = 0 ( = 1, 2,..., s), the submanfold M s nvarant and ξ s normal to M. Next, we assume that the nduced structure (ψ, V, v, g) s an almost paracontact Remannan structure. Then from (1.8) we have u (X)U = 0,
153 from whch u (X) = 0 ( = 1, 2,..., s) and from (1.9) we get α = 0. Thus M s nvarant and ξ s tangent to M. Theorem 2.3. In order that, n an almost paracontact Remannan manfold M wth a structure (ϕ, ξ, η, G), the submanfold M of M s nvarant, t s necessary and suffcent that the normal space T p (M) at every pont p M admts an orthonormal bass consstng of egenvectors of the matrx (ϕ). Proof. We assume that M s nvarant. 1.) When ξ s normal to M, at p M we consder an s-dmensonal vector space W and nvestgate the egenvalues of the (s, s)-matrx (λ j ). By means of (2.8) and (2.9), t s clear that the vector (α 1, α 2,..., α s ) of the vector space W s a unt egenvector of the matrx (λ j ) and ts egenvalue s equal to 0. Next, suppose that a vector (w 1, w 2,..., w s ) satsfyng α w = 0 s an egenvector and ts egnevalue s λ. Then we have λ j w j = λw. From j ths equaton, we get λ k λ j w j = λ λ k w, from whch ( λ k λ j )w j =,j j λ 2 w k. Usng (2.6), we have (δ kj α k α j )w j = λ 2 w k, that s, w k = λ 2 w k. j Then we get λ 2 = 1. Consequently, f by a sutable transformaton of the orthonormal bass {N 1, N 2,..., N s )} of T p (M), the matrx (λ j ) becomes a dagonal matrx, then the dagonal components λ 1, λ 2,..., λ s satsfy relatons λ 2 1 = λ 2 2 =... = λ 2 s 1, λ s = 0. In ths case, f we denote by { N 1, N 2,..., N s } the orthonormal bass of T p (M), then from (1.18) we have ϕ N l = λ l Nl. Therefore, N l (l = 1, 2,..., s) are egenvectors of the matrx (ϕ) and N s = ξ. 2.) When ξ s tangent to M, from (2.6) we have λ k λ j = δ kj. If we denote by (w 1, w 2,..., w s ) an egenvector of matrx (λ j ) and by λ ts egenvalue, then we have λ j w j = λw. Consequently, we have λ k λ j w j = j,j λ λ k w, that s, w k = λ 2 w k, from whch we get λ 2 = 1. Thus snce the egenvalues of (λ j ) are ±1, f by a sutable transformaton of the orthonormal bass of T p (M), {N 1, N 2,..., N s } becomes { N 1, N 2,..., N s }, then N 1, N 2,..., N s are egenvectors of matrx (ϕ). Conversely, f the orthonormal bass { N 1, N 2,..., N s } of T p (M) conssts of egenvectors of (ϕ) and these egenvalues λ 1, λ 2,..., λ s satsfy λ 2 1 = λ 2 2 =... = λ2 s 1 = 1, λ s = ±1 or 0, then we have ϕ N l = λ l Nl. Consequently, snce we have Ūl = 0, M s nvarant.
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