On doubly twisted product immersions
aa r X i v : . [ m a t h . DG ] J un ON DOUBLY TWISTED PRODUCT IMMERSIONS
ABDOUL SALAM DIALLO*, FORTUN ´E MASSAMBA**A
BSTRACT . Some basic geometric properties of doubly twisted productimmersions are established.
1. I
NTRODUCTION
One is often interested in the decomposition of mathematical objects as afirst step for a classification. A well-known theorem of de Rham gives suf-ficient conditions for a Riemannian manifold to be a Riemannian product.Also well-known is Moore’s theorem that gives a sufficient condition for anisometric immersion into a Euclidean space to split into a product immer-sion. An analogue for de Rham’s theorem for warped product metrics wasgiven by Hiepko [11]. Also, an analogue of Moore’s theorem for warpedproduct immersions was given by Nolker [14].Intrinsic and extrinsic invariants are very powerful to study submanifoldsof Riemannian manifolds, and to establish relationship between them isone of the must fundamental problems in submanifolds theory [4]. In thiscontext, Chen [2, 3] proved some basic inequalities for warped productsisometrically immersed in arbitrary Riemannian manifolds. Correspondinginequalities have been obtained for doubly warped product submanifoldsinto arbitrary Riemannian manifolds in [7, 15].In this paper, we consider doubly twisted products which are a gener-alization of singly warped products and doubly warped products. After apreliminaries section containing basic notions of Riemannian submanifoldstheory, warped product and doubly twisted products, we extend in Section 3an inequality of Chen involving the squared mean curvature vector obtainedin [3, 7] to doubly twisted products submanifolds. We finally end the paperby a question: May Theorems by Moore and Nolker (see [13] and [14] formore details) be extended to doubly twisted product immersions?2. P
RELIMINARIES
Background on submanifolds.
Let N and M be two differentiablemanifolds of dimensions n and m , respectively. We say that a differentiable Mathematics Subject Classification.
Primary 53B05; Secondary 53B20.
Key words and phrases.
Doubly twisted product, Immersion. map φ : N → M is an immersion if the differential φ ∗ ( p ) : T p N → T φ ( p ) M is injective for all p ∈ N . An immersion φ : N → M between two Rie-mannian manifolds with metrics g N and g M respectively, is called an iso-metric immersion if g M ( φ ∗ X, φ ∗ Y ) = g N ( X, Y ) , for every p ∈ N and forall vector fields X, Y tangent to N . One of the most fundamental prob-lems in submanifold theory is the immersibility of a Riemannian manifoldin a Euclidean space (or, more generally, in a space form). According tothe well-known theorem of Nash, every Riemannian n -manifold admitsan isometric immersion into the Euclidean space E n ( n +1)(3 n +11) / . In gen-eral, there exist enormously many isometric immersions from a Riemannianmanifold into Euclidean spaces if no restriction on the codimension is made[4]. For a submanifold of a Riemannian manifold there are associated sev-eral extrinsic invariants beside its intrinsic invariants. Among the extrinsicinvariants, the mean curvature vector and shape operator are the most fun-damental ones [4].Let φ : N → M be an isometric immersion of a Riemannian manifold ( N, g N ) into a Riemannian manifold ( M, g M ) . The formulas of Gauss andWeingarten are given respectively by ∇ X Y = ∇ X Y + h ( X, Y ) , ∇ X η = − A η X + D X η, (2.1)for all vectors fields X, Y tangent to N and η normal to N , where ∇ and ∇ denote the Levi-Civita connections on N and M , respectively; h is the sec-ond fundamental form; D the normal connection and A the shape operatorof φ . The shape operator and the second fundamental form are related by g M ( A η X, Y ) = g M ( h ( X, Y ) , η ) . Moreover, the mean curvature vector H of the submanifold N is defined by H = 1 n trace h = 1 n n X i =1 h ( e i , e i ) , where ( e , · · · , e n ) is a local orthonormal frame of the tangent bundle T N of N . The squared mean curvature is given by k H k = h H, H i . A submanifold N is said to be minimal in M if the mean curvature vectorof N in M vanishes identically. A submanifold N in a Riemannian man-ifold M is called totally geodesic if its second fundamental form vanishesidentically. It is said to be totally umbilical at p ∈ N if there is a realnumber λ such that h ( X, Y ) = λg M ( X, Y ) H for any X, Y ∈ T p N .2.2. Warped product manifolds.
Let ( N , g N ) and ( N , g N ) be two Rie-mannian manifolds of dimensions n and n , respectively; and let σ be apositive differentiable function on N . The warped product N × σ N is N DOUBLY TWISTED PRODUCT IMMERSIONS 3 defined to be the product manifold N × N equipped with the Riemannianmetric given by g N + σ g N . This notion was introduced by Bishop and O’Neill for constructing nega-tively curvature manifolds [1]. B. O’Neill in [16] discussed warped prod-ucts and explored curvature formulas of warped products in terms of cur-vatures of components of warped product. The warped products play someimportant role in differential geometry as well as in physics. Later this no-tion has been extended to a doubly warped product, a twisted product, adoubly twisted product and a multiply warped product manifolds.Let M × ρ M be a Riemannian warped product manifold and φ i : N i → M i , i = 1 , are isometric immersions between Riemannian manifolds. De-fine a positive function σ = ρ ◦ φ . Then the map φ : N × σ N → M × ρ M , given by φ ( x , x ) = ( φ ( x ) , φ ( x )) , is an isometric immersion, which iscalled a warped product immersion [3]. This notion appeared in severalrecent studies related to different geometric aspects. For examples, it ap-peared in the study of multi-rotation surfaces in [6], a decomposition prob-lem in [14], a geometric inequality and minimal immersion problem in [2]and also in the study done by M. Dajczer et al. [5]. Chen [2, 3] studied thefundamental geometry properties of warped product immersions.2.3. Twisted product manifolds.
Let ( N , g N ) and ( N , g N ) be two Rie-mannian manifolds of dimensions n and n , respectively; and let π : N × N → N and π : N × N → N be the canonical projections.Also, let σ : N × N → (0 , ∞ ) and σ : N × N → (0 , ∞ ) bepositive differentiable functions. The doubly twisted product of Riemann-ian manifolds ( N , g N ) and ( N , g N ) with twisting functions σ and σ is the product manifold N = N × N equipped with the metric tensor g N = σ g N ⊕ σ g N given by g N = σ π ∗ g N + σ π ∗ g N . We denote the Riemannian manifold ( N, g N ) by N × ( σ ,σ ) N . In partic-ular, if σ = 1 is constant, then N × ( σ ,σ ) N is called the twisted product of ( N , g N ) and ( N , g N ) with twisting function σ . Moreover, if σ de-pends only on N , then N × σ N is called warped product of ( N , g N ) and ( N, g N ) with the warping function σ .Ponge and Reckziegel [17] mentioned that the conformal change of aRiemannian metric can be interpreted as a twisted product, namely onewhere the first factor M consist of one point only. Therefore, formulas andassertions for doubly twisted products are applicable in many situations.For a vector field X on N , the lift of X to N × ( σ ,σ ) N is the vectorfield ˜ X whose value at each ( p, q ) is the lift of X p to ( p, q ) . Thus the lift ABDOUL SALAM DIALLO, FORTUN ´E MASSAMBA of X is the unique vector field on N × ( σ ,σ ) N that is π -related to X and π -related to the zero vector field on N . For a doubly twisted product N × ( σ ,σ ) N , let D i denote the distribution obtained from the vectorstangent to the horizontal lifts of N i .Denote by ∇ and ∇ the Levi-Civita connections on N × N equippedwith the doubly twisted product metric g N = σ g N ⊕ σ g N and with thedirect product metric g = g N + g N , respectively. We have the following. Proposition 2.1.
The Levi-Civita connections ∇ and ∇ are related by ∇ X Y = ∇ X Y + X (ln σ ) Y + Y (ln σ ) X − g N ( X, Y )grad(ln σ ) , (2.2) ∇ V W = ∇ V W + V (ln σ ) W + W (ln σ ) V − g N ( V, W )grad(ln σ ) , (2.3) ∇ X V = ∇ V X = X (ln σ ) V − V (ln σ ) X + V (ln σ ) X − X (ln σ ) V, (2.4) for any X, Y ∈ D and V, W ∈ D .Proof. We set ∇ X Y = ∇ X Y + θ X Y, for all X, Y ∈ Γ( T ( N × N )) .Then, it is easy to verify that θ is symmetric, that is, θ X Y = θ Y X . For any X, Y, Z ∈ Γ( T ( N × N )) , we have ( ∇ X g )( Y, Z ) = 0 , that is, X ( g ( Y, Z )) − g ( ∇ X Y, Z ) − g ( Y, ∇ X Z )= X ( g ( Y, Z )) − g ( ∇ X Y, Z ) − g ( θ X Y, Z ) − g ( Y, ∇ X Z ) − g ( Y, θ X Z ) . Since g N = σ g N , so for all X, Y, Z ∈ Γ( T N ) , we have X ( σ ) g N ( Y, Z ) + σ X ( g N ( Y, Z )) − σ (cid:2) g N ( ∇ X Y, Z ) + g N ( θ X Y, Z ) + g N ( Y, ∇ X Z ) + g N ( Y, θ X Z ) (cid:3) = X ( σ ) g N ( Y, Z ) − σ [ g N ( θ X Y, Z ) + g N ( Y, θ X Z )] . That is, one obtains g N ( θ X Y, Z ) + g N ( Y, θ X Z ) = 2 X (ln σ ) g N ( Y, Z ) . (2.5)A circular permutation in (2.5) gives g N ( θ Y Z, X ) + g N ( Z, θ Y X ) = 2 Y (ln σ ) g N ( Z, X ) , (2.6) g N ( θ Z X, Y ) + g N ( X, θ Z Y ) = 2 Z (ln σ ) g N ( X, Y ) . (2.7)Putting the pieces above using the operation (2.5) + (2.6) - (2.7), we have g N ( θ X Y, Z ) = X (ln σ ) g ( Y, Z ) + Y (ln σ ) g ( X, Z ) − g N ( X, Y ) g N (grad ln σ , Z ) . N DOUBLY TWISTED PRODUCT IMMERSIONS 5
Thus θ X Y = X (ln σ ) Y + Y (ln σ ) X − g N ( X, Y )grad ln σ , which provesthe relation (2.2). Similarly, we can obtain the relations (2.3) and (2.4). (cid:3) In [17], Ponge and Reckziegel gave a characterization of a twisted prod-uct pseudo-Riemannian manifold in terms of distributions defined on themanifolds. Recently, in [8], Fernadez-Lopez et al. gave a condition fora twisted product manifold to be a warped product manifold by using theRicci tensor of the manifold. Similar characterizations were given by Kazanand Sahin in [12] by imposing certain conditions on the Weyl conformalcurvature tensor and the Weyl projective tensor of the manifold.3. T
HE MAIN RESULTS
Let M × ( ρ ,ρ ) M be a doubly twisted product of two Riemannian mani-folds and let ( φ , φ ) : N × N → M × M be a direct product immersion.Define a positive function σ i , i = 1 , on N × N by σ i = ρ i ◦ ( φ × φ ) .The map φ : N × ( σ ,σ ) N → M × ( ρ ,ρ ) M , given by φ ( x , x ) =( φ ( x ) , φ ( x )) is an isometric immersion, which is called a doubly twistedproduct isometric immersion .Let M × ( ρ ,ρ ) M be a doubly twisted product of two Riemannian man-ifolds M and M equipped with Riemannian metrics g M and g M , re-spectively, where ρ and ρ are two positive smooth functions define on M × M . Denote by D and D the distributions obtained from vectorstangent to the horizontal lifts of M and M , respectively. Denote by ∇ and ∇ the Levi-Civita connections of M × M equipped with the doublytwisted product metric g M = ρ g M + ρ g M and with the direct prod-uct metric g = g M + g M , respectively. Then, from Proposition 2.1, theLevi-Civita connections ∇ and ∇ are related by ∇ X Y = ∇ X Y + X (ln ρ ) Y + Y (ln ρ ) X − g M ( X, Y )grad(ln ρ ) , (3.1) ∇ V W = ∇ V W + V (ln ρ ) W + W (ln ρ ) V − g M ( V, W )grad(ln ρ ) , (3.2) ∇ X V = ∇ V X = X (ln ρ ) V − V (ln ρ ) X + V (ln ρ ) X − X (ln ρ ) V, (3.3)for all X, Y ∈ D and V, W ∈ D , (cf. [14, 17]).Let h φ denote the second fundamental form of a doubly twisted productimmersion φ : N × ( σ ,σ ) N → M × ( ρ ,ρ ) M and let h denote thesecond fundamental form of the corresponding direct product immersion ( φ , φ ) : N × N → M × M , respectively. By applying (3.1), (3.2), ABDOUL SALAM DIALLO, FORTUN ´E MASSAMBA (3.3) and Gauss formula, we obtain h φ ( X, Y ) = h ( X, Y ) + X (ln ρ ) Y + Y (ln ρ ) X − g M ( X, Y ) D ln ρ , (3.4) h φ ( V, W ) = h ( V, W ) + V (ln ρ ) W + W (ln ρ ) V − g M ( V, W ) D ln ρ , (3.5) h φ ( X, V ) = 0 , (3.6)for all X, Y ∈ D and V, W ∈ D .The restriction of h to D and to D are the second fundamental formsof φ : N → M and φ : N → M , respectively. Hence, h ( X, Y ) and h ( V, W ) are mutually orthogonal for X, Y ∈ D and V, W ∈ D .Let φ : N × ( σ ,σ ) N → M be an isometric immersion of a doublytwisted product N × ( σ ,σ ) N into a Riemannian manifold M of constantsectional curvature c . Let h and h denote the restrictions of the secondfundamental form h φ to D and D respectively. Then φ is called N i -totallygeodesic if the partial second fundamental forms h i , i = 1 , vanish identi-cally [4]. Theorem 3.1.
Let φ : N × ( σ ,σ ) N → M × ( ρ ,ρ ) M be a doubly twistedproduct immersion with dim N = n and dim N = n and Ψ( ρ , ρ ) = 2 n X i,j =1 g N (cid:16) h ( e i , e j ) , e i (ln ρ ) e j + e j (ln ρ ) e i (cid:17) + n X i,j =1 g N (cid:16) e i (ln ρ ) e j + e j (ln ρ ) e i , e i (ln ρ ) e j + e j (ln ρ ) e i (cid:17) − n X i,j =1 g N (cid:16) h ( e i , e j ) + e i (ln ρ ) e j + e j (ln ρ ) e i , g N ( e i , e j ) D ln ρ (cid:17) + 2 n + n X α,β = n +1 g N (cid:16) h ( e α , e β ) , e α (ln ρ ) e β + e β (ln ρ ) e α (cid:17) + n + n X α,β = n +1 g N (cid:16) e α (ln ρ ) e β + e β (ln ρ ) e α , e α (ln ρ ) e β + e β (ln ρ ) e α (cid:17) − n + n X α,β = n +1 g N (cid:16) h ( e α , e β ) + e α (ln ρ ) e β + e β (ln ρ ) e α , g N ( e α , e β ) D ln ρ (cid:17) , Then the following statements hold:
N DOUBLY TWISTED PRODUCT IMMERSIONS 7 (i)
The squared norm of the second fundamental form of φ satisfies k h φ k ≥ n k D (ln ρ ) k + n k D (ln ρ ) k + Ψ( ρ , ρ ) . (ii) φ is N -totally geodesic if and only φ : N → M is totally geo-desic and X (ln ρ ) Y + Y (ln ρ ) X = g M ( X, Y ) D ln ρ . (iii) φ is N -totally geodesic if and only φ : N → M is totally geo-desic and V (ln ρ ) W + W (ln ρ ) V = g M ( V, W ) D ln ρ . (iv) φ is a totally geodesic immersion if and only if φ is both N i -totallygeodesics.Proof. (i) is proven as follows. Let e i ∈ D i , i = 1 , · · · , n and e α ∈ D α , α = n + 1 , · · · , n + n be orthonormal frame fields of N × ( σ ,σ ) N .Then we have k h φ k = n + n X a,b =1 g N (cid:0) h φ ( e a , e b ) , h φ ( e a , e b ) (cid:1) = n X i,j =1 g N (cid:0) h ( e i , e j ) , h ( e i , e j ) (cid:1) + n + n X α,β = n +1 g N (cid:0) h ( e α , e β ) , h ( e α , e β ) (cid:1) + n X i,j =1 g N ( g ( e i , e j ) D ln ρ , g ( e i , e j ) D ln ρ )+ n + n X α,β = n +1 g N ( g ( e α , e β ) D ln ρ , g ( e α , e β ) D ln ρ )+ Ψ( ρ , ρ ) . If φ : N × ( σ ,σ ) N → M is a N i -totally geodesic immersion, then itfollows from (3.4) and (3.5) that h ( X, Y ) + X (ln ρ ) Y + Y (ln ρ ) X − g M ( X, Y ) D ln ρ , h ( V, W ) + V (ln ρ ) W + W (ln ρ ) V − g M ( V, W ) D ln ρ . Since h ( X, Y ) , D ln ρ and h ( V, W ) , D ln ρ are orthogonal respectively,we have h ( X, Y ) = 0 , h ( V, W ) = 0 , that is, φ and φ are totally geodesic, and X (ln ρ ) Y + Y (ln ρ ) X − g M ( X, Y ) D ln ρ , V (ln ρ ) W + W (ln ρ ) V − g M ( V, W ) D ln ρ . ABDOUL SALAM DIALLO, FORTUN ´E MASSAMBA
Conversely, if φ and φ are totally geodesic and X (ln ρ ) Y + Y (ln ρ ) X − g M ( X, Y ) D ln ρ , V (ln ρ ) W + W (ln ρ ) V − g M ( V, W ) D ln ρ . it follows from (3.4) and (3.5) that h φ ( X, Y ) = 0 and h φ ( V, W ) = 0 for
X, Y ∈ D and V, W ∈ D . This completes the proof of (ii) and (iii).The statement (iv) follows from (ii) and (iii). (cid:3) For a doubly warped product immersion, that is, when σ (respectively σ depend only on N (respectively N ) and ρ (respectively ρ depend onlyon M (respectively M ); then we have the following result by Faghfouriand Majidi [7]. Corollary 3.2. [7, Theorem 1]
Let φ : ( φ , φ ) : N × ( σ ,σ ) N → M × ( ρ ,ρ ) M be a doubly warped product immersion between two doubly warpedproduct manifolds. Then the following statements are true: (i) φ is mixed totally geodesic. (ii) The squared norm of the second fundamental form of φ satisfies (cid:13)(cid:13) h φ (cid:13)(cid:13) ≥ n k D ln ρ k + n k D ln ρ k with equality holding if and only if φ : N → M and φ : N → M are both totally geodesic immersions. (iii) φ is N -totally geodesic if and only if φ : N → M is totallygeodesic and D ln ρ = 0 . (iv) φ is N -totally geodesic if and only if φ : N → M is totallygeodesic and D ln ρ = 0 . (v) φ is a totally geodesic immersion if and only if φ is both N -totallygeodesic and N -totally geodesic. In the case of warped product immersion, we have the following result ofChen.
Corollary 3.3. [3, Theorem 3]
Let φ : ( φ , φ ) : N × σ N → M × ρ M bea warped product immersion between two warped product manifolds. Thenthe following statements are true: (i) φ is mixed totally geodesic. (ii) The squared norm of the second fundamental form of φ satisfies (cid:13)(cid:13) h φ (cid:13)(cid:13) ≥ n k D ln ρ k with equality holding if and only if φ : N → M and φ : N → M are both totally geodesic immersions. N DOUBLY TWISTED PRODUCT IMMERSIONS 9 (iii) φ is N -totally geodesic if and only if φ : N → M is totallygeodesic. (iv) φ is N -totally geodesic if and only if φ : N → M is totallygeodesic and ( ∇ ln ρ ) | N = ∇ ln f holds, that is, the restriction ofthe gradient of ln ρ to N is the gradient of ln f , or equivalently, D ln ρ = 0 . (v) φ is a totally geodesic immersion if and only if φ is both N -totallygeodesic and N -totally geodesic. Let φ : N × ( σ ,σ ) N → M be an isometric immersion of a doublytwisted product N × ( σ ,σ ) N into a Riemannian manifold M with constantsectional curvature c . Denote by trace h and trace h the trace of h i , i =1 , restricted to N and N , respectively, that is trace h = n X α =1 h ( e α , e α ) , and trace h = n + n X t = n +1 h ( e t , e t ) for some orthonormal frame fields e , · · · , e n and e n +1 , · · · , e n + n of D and D , respectively. The partial mean curvature vectors H i is defined by H = 1 n trace h and H = 1 n trace h . An immersion φ : N × ( σ ,σ ) N → M is called N i -minimal if the partialmean curvatures H i , i = 1 , vanish identically [4].In the sequel, we need the following lemma. Lemma 3.4. [10]
Let f be a smooth function that take its arguments fromsome product space N × N . Then, normal component Df of the gradientof f on N × N can be decomposed as Df = D f + D f, where D i f are normal components of the gradients of f on N i , i = 1 , . Theorem 3.5.
Let φ : N × ( σ ,σ ) N → M × ( ρ ,ρ ) M be a doubly twistedproduct immersion between two doubly twisted product manifolds. Then thefollowing statements are true: (i) φ is N -minimal if and only φ : N → M is a minimal isometric, n σ D ln ρ = 2 n X i =1 e i (ln ρ ) e i and D ln ρ = 0 . (ii) φ is N -minimal if and only φ : N → M is a minimal isometric, n σ D ln ρ = 2 n X α =1 e α (ln ρ ) e α and D ln ρ = 0 . (iii) φ is a minimal immersion if and only if the mean curvature vectorsof φ and φ are given by n − n σ D ln ρ + σ D ln ρ − n − n X a =1 e a (ln ρ ) e a , and n n − σ D ln ρ + σ D ln ρ − n − n X α =1 e α (ln ρ ) e α , respectively.Proof. Let e i ∈ D , i = 1 , · · · , n and e α ∈ D , α = n + 1 , · · · , n + n be orthonormal frame fields of N × ( σ ,σ ) N . Recall that the partial meancurvature H i is defined by H i = 1 n i trace h i . (i) If φ : N × ( σ ,σ ) N → M × ( ρ ,ρ ) M is N -minimal, then it followsfrom equation (4.4) that H = 1 n n X i =1 (cid:0) h ( e i , e i ) + 2 e i (ln ρ ) e i − g M ( e i , e i ) D ln ρ (cid:1) = 0 . From Lemma 3.4, we have D ln ρ = D ln ρ + D ln ρ and n n X i =1 h ( e i , e i ) + 2 1 n n X i =1 e i (ln ρ ) e i − D ln ρ − D ln ρ = 0 . That is, h n + 2 1 n n X i =1 e i (ln ρ ) e i − D ln ρ − D ln ρ . Since D ln ρ and trace h are orthogonal, we have φ is minimal immer-sion, n D ln ρ = 2 n X i =1 e i (ln ρ ) e i , and D ln ρ = 0 . (ii) is similar to (i).Now we prove (iii). If φ : N × ( σ ,σ ) N → M × ( ρ ,ρ ) M is minimalimmersion, then trace h = 0 . By applying equations (4.4) and (4.5), we get n X a =1 (cid:16) h ( e a , e a ) + 2 e a (ln ρ ) e a − n σ D ln ρ − n σ D ln ρ (cid:17) + n X α =1 (cid:16) h ( e α , e α ) + 2 e α (ln ρ ) e α − n σ D ln ρ − n σ D ln ρ (cid:17) , N DOUBLY TWISTED PRODUCT IMMERSIONS 11 which implies that h + 2 n X a =1 e a (ln ρ ) e a − n σ D ln ρ − n σ D ln ρ + trace h + 2 n X a =1 e α (ln ρ ) e α − n σ D ln ρ − n σ D ln ρ . Since trace h , n X a =1 e a (ln ρ ) e a , D ln ρ and D ln ρ are tangent to the firstfactor M , and trace h , n X a =1 e α (ln ρ ) e α , D ln ρ and D ln ρ are tangentto the second factor M , then n trace h = n − n σ D ln ρ + σ D ln ρ − n n X a =1 e a (ln ρ ) e a and n trace h = σ D ln ρ + n − n σ D ln ρ − n n X α =1 e α (ln ρ ) e α which complete the proof. (cid:3)
4. O
N DOUBLY TWISTED PRODUCT IMMERSIONS
A basic problem in the theory of submanifold is to provide conditionsthat imply that an isometric immersion of a product manifold must be aproduct of isometric immersions. The first contribution to this problem wasgiven by Moore [13]. The latter showed the following result:
Theorem 4.1. [13]
Let φ : N → R n be an isometric immersion of a Rie-mannian product N = N × N × · · · × N k of connected Riemannianmanifolds into the Euclidean space R n . Then φ is a mixed totally geodesicimmersion if and only if φ is a product immersion, that is, there exists anisometry ψ : M × M ×· · ·× M k → R n , and there are isometric immersions φ i : N i → M i , i = 1 , . . . , k , such that φ = ψ ◦ ( φ × φ × · · · × φ k ) . In [9], Ferus used Moore’s Theorem to factorize isometric immersionsinto R n with parallel second fundamental form. This was an important steptowards their classification. In [6], Dillen and Nolker classified normallyflat semi-parallel (a condition slightly weaker than that of a parallel secondfundamental form) submanifolds of a warped product manifolds. Nolkerextended Moore’s result to isometric immersions of multiply warped prod-ucts. Theorem 4.2. [14]
Let φ : N × σ N ×· · ·× σ k N k → R m ( c ) be an isometricimmersion of a multiply warped product into a complete simply-connectedreal space form R m ( c ) of constant curvature c . Then φ is mixed totallygeodesic immersion if and only if there exists an isometry ψ : M × ρ M ×· · · × ρ k M k → R m ( c ) , where M is an open subset of a standard space and M , . . . , M k are standard spaces, and there exist isometric immersions φ i : N i → M i , i = 1 , · · · , k such that σ i = ρ i ◦ φ , for i = 2 , · · · , k and φ = ψ ◦ ( φ × φ × · · · × φ k ) . In the paper [5], Dajczer and Vlachos gave a condition under which anisometric immersion of a warped product of manifolds into a space formmust be a warped product of isometric immersions.Let N × ( σ ,σ ) N be a doubly twisted product of two Riemannian mani-folds N and N equipped with Riemannian metrics g N and g N of dimen-sions n and n , respectively, where σ and σ are two positive smooth func-tions defined on N × N . Denote by D and D the distributions obtainedfrom the vectors tangent to N and N , respectively, (or more precisely,from vectors tangent to the horizontal lifts of N and N , respectively).Let φ : N × ( σ ,σ ) N → M be an isometric immersion of a doublytwisted product N × ( σ ,σ ) N into a Riemannian manifold M . Denote by h the second fundamental form of φ . Then φ is called mixed totally geodesic if its second fundamental form h satisfies h ( X, V ) = 0 , for any X ∈ D and V ∈ D .We finally end this section by the following problem: Let φ : N × ( σ ,σ ) N → R m be an isometric immersion of a connected doubly twisted product N = N × ( σ ,σ ) N into a real space form R m ( c ) of constant curvature c . Now, if φ is mixed totally geodesic, is there an isometric immersion ψ : M × ( ρ ,ρ ) M → G from a doubly twisted product M × ( ρ ,ρ ) M ontoan open, dense subset G ⊂ R m and a direct product isometric immersion ( φ , φ ) : N × N → M × M such that σ i = ρ i ◦ ( φ × φ ) for i = 1 , and φ = ψ ◦ ( φ × φ ) ? A CKNOWLEDGMENTS
The authors would like to thank Professor Mukut Mani Tripathi (BanarasHindu University, India) for his many valuable suggestions and comments.ASD is thankful to the University of KwaZulu-Natal for financial support.R
EFERENCES [1] R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math.Soc. 145 (1969), 1-49.[2] B. Y. Chen, On isometric minimal immersions from warped products into real spaceforms, Proc. Edinburgh Math. Soc. 45 (2002), 579-587.
N DOUBLY TWISTED PRODUCT IMMERSIONS 13 [3] B. Y. Chen, On warped product immersions, J. Geom. 82 (2005), 36-49.[4] B. Y. Chen, Pseudo-Riemannian geometry, δ -Invariants and Applications, WorldScientific Publishing Company, Singapore (2011).[5] M. Dajczer and T. Vlachos, Isometric immersions of warped product, Proc. Amer.Math. Soc. 141, (2013), no. 5, 1795-1803.[6] F. Dillen and S. Nolker, Semi-parallelity, multi-rotation surfaces and their helixproperty, J. Reine Angew. Math. 435 (1993), 33-63.[7] M. Faghfouri and A. Majidi, On doubly warped product immersions, J. Geom. 106(2015), 243-254.[8] M. Fernadez-Lopez, E. Garcia-Rio, D. Kupeli and B. Unal, A curvature conditionfor a twisted product to be a warped product, Manuscripta Math. 106 (2001), 213-217.[9] D. Ferus, Produkt-Zerlegung von Immersionen mit paralleler zweiter Fundamental-form, Math. Ann. 211 (1974), 1-5.[10] M. Grosser, M. Kunzinger, M. Oberguggenberger and R. Steinbauer, Geometrictheory of generalized functions with applications to general relativity, 537 KluwerAcademic Publishers, Dordrecht, 2001.[11] S. Hiepko, Eine innere Kennzeichnung der verzerrten Produkte, Math. Ann. 241(1979), no. 3, 209-215.[12] S. Kazan and B. Sahin, Characterizations of twisted product manifolds to be warpedproduct manifolds, Acta Math. Univ. Comenian. (N.S.) 82 (2013), no. 2, 253-263.[13] J. D. Moore, Isometric immersions of Riemannian products, J. Differential Geom. 5(1971), 159-168.[14] S. Nolker, Isometric immersions of warped products, Differ. Geom. Appl. 6 (1996),1-30.[15] A. Olteanu, A general inequality for doubly warped product submanifolds, Math. J.Okayama Univ. 52 (2010), 133-142.[16] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, AcademicPress, London, 1983.[17] R. Ponge and H. Reckziegel, Twisted products in pseudo-Riemannian geometry,Geom. Dedicata 48 (1993), no. 1, 15-25.* S CHOOL OF M ATHEMATICS , S
TATISTICS AND C OMPUTER S CIENCE U NIVERSITY OF K WA Z ULU -N ATAL P RIVATE B AG X01, S
COTTSVILLE
OUTH A FRICAAND U NIVERSIT ´ E A LIOUNE D IOP DE B AMBEY
UFR SATIC, D ´
EPARTEMENT DE M ATH ´ EMATIQUES
B. P. 30, B
AMBEY , S ´ EN ´ EGAL
E-mail address : [email protected], [email protected] ** S
CHOOL OF M ATHEMATICS , S
TATISTICS AND C OMPUTER S CIENCE U NIVERSITY OF K WA Z ULU -N ATAL P RIVATE B AG X01, S
COTTSVILLE
OUTH A FRICA
E-mail address ::