On the estimates of warping functions on isometric immersions
aa r X i v : . [ m a t h . DG ] F e b ON THE ESTIMATES OF WARPING FUNCTIONS ONISOMETRIC IMMERSIONS
KWANG-SOON PARK
Abstract.
Using the results of [11], we get some estimates of warping func-tions for isometric immersions by changing the target manifolds by some typesof Riemannian manifolds: constant space forms and Hermitian symmetricspaces. And we deal with equality cases and obtain their applications. Fi-nally, we give some open problems. Introduction
Let (
B, g B ) and ( F, g F ) be Riemannian manifolds. Given a warped productmanifold M = B × f F with a warping function f (See [11]), we can consider anisometric immersion ψ : M ( M , g ), where (
M , g ) is a Riemannian manifold.In 2018, B. Y. Chen [5] proposed two Fundamental Questions on the isometricimmersion ψ : M ( M , g ) and gave some recent results on these problems where(
M , g ) is a K¨ahler manifold.In 2014, as a generalization of Chen’s works ([3],[4]), the author [11] obtainedtwo inequalities, which give the upper bound and the lower bound of the func-tion △ ff . Replacing the Riemannian manifold ( M , g ) with several types of Rie-mannian manifolds (i.e., real space forms, complex space forms, quaternionic spaceforms, Sasakian space forms, Kenmotsu space forms, Hermitian symmetric spaces:complex two-plane Grassmannians, complex hyperbolic two-plane Grassmannians,complex quadrics), we will obtain the upper bounds and the lower bounds of thefunctions △ ff . And by using these results, we will get some equality cases of theserelations and obtain their applications.We also know that warped product manifolds take an important position indifferential geometry and in physics, in particular in general relativity. And Nash’sresult [9] implies that each warped product manifold can be isometrically embeddedin a Euclidean space.The paper is organized as follows. In section 2 we remind some notions, whichwill be used in the following sections. In section 3 we estimate the upper boundsand the lower bounds of the functions △ ff for constant space forms ( M , g ) andhave some equality cases and their applications. In section 4 we do the works forHermitian symmetric spaces (
M , g ). In section 5 we give some open problems.2.
Preliminaries
In this section we recall some notions, which will be used in the following sections.
Mathematics Subject Classification.
Key words and phrases. warping function, warped product, isometric immersion, eigenvalue.
Let (
M , g ) be an n -dimensional Riemannian manifold and let M be an m -dimensional submanifold of ( M , g ). We denote by ∇ and ∇ the Levi-Civita con-nections of M and M , respectively.Then we get the Gauss formula and the
Weingarten formula ∇ X Y = ∇ X Y + h ( X, Y ) , (2.1) ∇ X N = − A N X + D X N, (2.2)respectively, for tangent vector fields X, Y ∈ Γ( T M ) and a normal vector field N ∈ Γ( T M ⊥ ), where h , A , D denote the second fundamental form , the shapeoperator , the normal connection of M in M , respectively.Then we know(2.3) g ( A N X, Y ) = g ( h ( X, Y ) , N ) . Fix a local orthonormal frame { v , · · · , v n } of T M with v i ∈ Γ( T M ), 1 ≤ i ≤ m and v α ∈ Γ( T M ⊥ ), m + 1 ≤ α ≤ n . We define the mean curvature vector field H ,the squared mean curvature H , the squared norm || h || of the second fundamentalform h as follows: H = 1 m trace h = 1 m m X i =1 h ( v i , v i ) , (2.4) H = g ( H, H ) , (2.5) || h || = m X i,j =1 g ( h ( v i , v j ) , h ( v i , v j )) . (2.6)We call the submanifold M ⊂ ( M , g ) totally geodesic if the second fundamentalform h vanishes identically. Denote by R , R the Riemannian curvature tensors of M , M , respectively.Let K ( X ∧ Y ) := g ( R ( X, Y ) Y, X ) g ( X, X ) g ( Y, Y ) − g ( X, Y ) ,K ( X ∧ Y ) := g ( R ( X, Y ) Y, X ) g ( X, X ) g ( Y, Y ) − g ( X, Y ) for X, Y ∈ Γ( T M ), where g denotes the induced metric on M of ( M , g ). i.e., givena plane V ⊂ T p M , p ∈ M , spanned by vectors X, Y ∈ T p M , K ( V ) = K ( X ∧ Y )and K ( V ) = K ( X ∧ Y ) denote the sectional curvatures of a plane V in M and in M , respectively.Let (inf K )( p ) := inf { K ( V ) | V ⊂ T p M, dim V = 2 } , (2.7) (sup K )( p ) := sup { K ( V ) | V ⊂ T p M, dim V = 2 } . (2.8)Let R ( X, Y, Z, W ) := g ( R ( X, Y ) Z, W ) for
X, Y, Z, W ∈ Γ( T M ).Given a C ∞ − function f ∈ C ∞ ( M ), we define the Laplacian △ f of f by △ f := m X i =1 (( ∇ v i v i ) f − v i f ) . Let (
B, g B ) and ( F, g F ) be Riemannian manifolds. N THE ESTIMATES OF WARPING FUNCTIONS 3
Throughout this paper, we will denote by (
M, g ) := ( B × f F, g B + f g F ) thewarped product manifold of Riemannian manifolds ( B, g B ) and ( F, g F ) with thewarping function f : B R + (See [11]).By Theorem 3.1, Theorem 3.4, and their proofs of [11], we have Lemma 2.1.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold andlet ( M , g ) be a Riemannian manifold. Let ψ : ( M, g ) ( M , g ) be an isometricimmersion. Then we get (2.9) m m m − H − m || h || + m inf K ≤ △ ff ≤ m m H + m sup K, where m = dim B and m = dim F with m = m + m . Constant space forms
In this section, we will estimate the functions △ ff for isometric immersions ψ :( M, g ) = ( B × f F, g B + f g F ) ( M , g ) with constant space forms (
M , g ). We alsodeal with equality cases and obtain their applications.Using Lemma 2.1, we obtain
Theorem 3.1.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , g ) a real space form of constant sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Then we have (3.1) m m m − H − m || h || + m c ≤ △ ff ≤ m m H + m c, where m = dim B and m = dim F with m = m + m .Proof. We know that the Riemannian curvature tensor R [8] of ( M , g ) is given by(3.2) R ( X, Y ) Z = c ( g ( Y, Z ) X − g ( X, Z ) Y )for X, Y, Z ∈ Γ( T M ). Since inf K = sup K = c , by Lemma 2.1, we get the result. (cid:3) Then we easily obtain
Corollary 3.2.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , g ) a real space form of constant sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume that the manifold ( M, g ) is a totally geodesic submanifold of ( M , g ) . Then we get m c ≤ △ ff ≤ m c . Remark . Let (
M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , g ) a real space form of constant sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume that the manifold (
M, g )is a totally geodesic submanifold of (
M , g ).Then the warping function f is an eigen-function with eigenvalue m c .In particular, if c = 0 (i.e., ( M , g ) is a Euclidean space E n .), then the warpingfunction f is a harmonic function. KWANG-SOON PARK
Lemma 3.4.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , g ) a real space form of constant sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a totally geodesic submanifold ( M, g ) of ( M , g ) such thateither the warping function f is not an eigen-function or the eigenvalue of f is notequal to m c . Theorem 3.5.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , g, J ) a complex space form of constant holomorphic sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Then we have m m m − H − m || h || + m c ≤ △ ff ≤ m m H + m c, c ≥ , (3.3) m m m − H − m || h || + m c ≤ △ ff ≤ m m H + m c , c < , (3.4) where m = dim B and m = dim F with m = m + m .Proof. We see that the Riemannian curvature tensor R [8] of ( M , g ) is given by R ( X, Y ) Z (3.5) = c g ( Y, Z ) X − g ( X, Z ) Y + g ( JY, Z ) JX − g ( JX, Z ) JY − g ( JX, Y ) JZ )for X, Y, Z ∈ Γ( T M ). Given orthonormal vectors
X, Y ∈ T p M , p ∈ M , we get(3.6) K ( X ∧ Y ) = R ( X, Y, Y, X ) = c g ( JX, Y ) )so that since 0 ≤ | g ( JX, Y ) | ≤
1, we easily obtain c ≤ K ( X ∧ Y ) ≤ c, c ≥ ,c ≤ K ( X ∧ Y ) ≤ c , c < . From Lemma 2.1, the result follows. (cid:3)
Corollary 3.6.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , g, J ) a complex space form of constant holomorphic sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume thatthe manifold ( M, g ) is a totally geodesic totally real submanifold of ( M , g ) (i.e., J ( T M ) ⊂ T M ⊥ ).Then we have m c ≤ △ ff ≤ m c . Proof.
By Lemma 2.1 and (3.6), we obtain the result. (cid:3)
Remark . Let (
M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , g, J ) a complex space form of constant holomorphic sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume that themanifold (
M, g ) is a totally geodesic totally real submanifold of (
M , g ).Then the warping function f is an eigen-function with eigenvalue m c . N THE ESTIMATES OF WARPING FUNCTIONS 5
Lemma 3.8.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , g, J ) a complex space form of constant holomorphic sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a totally geodesic totally real submanifold ( M, g ) of ( M , g ) such that either the warping function f is not an eigen-function or the eigenvalueof f is not equal to m c . Corollary 3.9.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , g, J ) a complex space form of constant holomorphic sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume that themanifold ( M, g ) is a 2-dimensional totally geodesic complex submanifold of ( M , g ) (i.e., J ( T M ) =
T M ).Then we have m c ≤ △ ff ≤ m c. Remark . Let (
M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , g, J ) a complex space form of constant holomorphic sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume that themanifold (
M, g ) is a 2-dimensional totally geodesic complex submanifold of (
M , g ).Then the warping function f is an eigen-function with eigenvalue m c . Lemma 3.11.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , g, J ) a complex space form of constant holomorphic sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a 2-dimensional totally geodesic complex submanifold ( M, g ) of ( M , g ) such that either the warping function f is not an eigen-function or theeigenvalue of f is not equal to m c . Theorem 3.12.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = ( M ( c ) , E, g ) a quaternionic space form of constant quaternionic sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Then we obtain m m m − H − m || h || + m c ≤ △ ff ≤ m m H + m c, c ≥ , (3.7) m m m − H − m || h || + m c ≤ △ ff ≤ m m H + m c , c < , (3.8) where m = dim B and m = dim F with m = m + m .Proof. We know that the Riemannian curvature tensor R [6] of ( M , g ) is given by R ( X, Y ) Z = c g ( Y, Z ) X − g ( X, Z ) Y (3.9) + X α =1 ( g ( J α Y, Z ) J α X − g ( J α X, Z ) J α Y − g ( J α X, Y ) J α Z ))for X, Y, Z ∈ Γ( T M ). Given orthonormal vectors
X, Y ∈ T p M , p ∈ M , we have(3.10) K ( X ∧ Y ) = R ( X, Y, Y, X ) = c X α =1 g ( J α X, Y ) ) . KWANG-SOON PARK
Since { J X, J X, J X } is orthonormal, we get 0 ≤ P α =1 g ( J α X, Y ) ≤ | Y | = 1so that c ≤ K ( X ∧ Y ) ≤ c, c ≥ ,c ≤ K ( X ∧ Y ) ≤ c , c < . From Lemma 2.1, we obtain the result. (cid:3)
Corollary 3.13.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , E, g ) a quaternionic space form of constant quaternionic sec-tional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assumethat the manifold ( M, g ) is a totally geodesic totally real submanifold of ( M , g ) (i.e., J α ( T M ) ⊂ T M ⊥ , ∀ α ∈ { , , } ).Then we have m c ≤ △ ff ≤ m c . Proof.
By Lemma 2.1 and (3.10), we obtain the result. (cid:3)
Lemma 3.14.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = ( M ( c ) , E, g ) a quaternionic space form of constant quaternionic sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a totally geodesic totally real submanifold ( M, g ) of ( M , g ) such that either the warping function f is not an eigen-function or the eigenvalueof f is not equal to m c . Corollary 3.15.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = ( M ( c ) , E, g ) a quaternionic space form of constant quaternionic sec-tional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assumethat the manifold ( M, g ) is a 4-dimensional totally geodesic quaternionic submani-fold of ( M , g ) (i.e., J α ( T M ) =
T M , ∀ α ∈ { , , } ).Then we have m c ≤ △ ff ≤ m c. Lemma 3.16.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = ( M ( c ) , E, g ) a quaternionic space form of constant quaternionic sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a 4-dimensional totally geodesic quaternionic submanifold ( M, g ) of ( M , g ) such that either the warping function f is not an eigen-functionor the eigenvalue of f is not equal to m c . Theorem 3.17.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Sasakian space form of constant φ -sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Then we obtain m m m − H − m || h || + m ≤ △ ff ≤ m m H + m c, c ≥ , (3.11) m m m − H − m || h || + m c ≤ △ ff ≤ m m H + m , c < , (3.12) where m = dim B and m = dim F with m = m + m . N THE ESTIMATES OF WARPING FUNCTIONS 7
Proof.
We see that the Riemannian curvature tensor R [10] of ( M , g ) is given by R ( X, Y ) Z = c + 34 ( g ( Y, Z ) X − g ( X, Z ) Y )(3.13) + c −
14 ( η ( X ) η ( Z ) Y − η ( Y ) η ( Z ) X + η ( Y ) g ( X, Z ) ξ − η ( X ) g ( Y, Z ) ξ + g ( φY, Z ) φX − g ( φX, Z ) φY − g ( φX, Y ) φZ )for X, Y, Z ∈ Γ( T M ). Given orthonormal vectors
X, Y ∈ T p M , p ∈ M , we have(3.14) K ( X ∧ Y ) = R ( X, Y, Y, X ) = c + 34 + c −
14 ( − η ( Y ) − η ( X ) +3 g ( φX, Y ) ) . If ξ ∈ Span(
X, Y ), then − η ( Y ) − η ( X ) + 3 g ( φX, Y ) = −
1. If Y = φX and η ( X ) = 0, then − η ( Y ) − η ( X ) + 3 g ( φX, Y ) = 3. Hence we get − ≤ − η ( Y ) − η ( X ) + 3 g ( φX, Y ) ≤ ≤ K ( X ∧ Y ) ≤ c, c ≥ ,c ≤ K ( X ∧ Y ) ≤ , c < . From Lemma 2.1, the result follows. (cid:3)
Corollary 3.18.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product mani-fold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Sasakian space form of constant φ -sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume thatthe manifold ( M, g ) is a totally geodesic φ -totally real submanifold of ( M , g ) with ξ ∈ Γ( T M ⊥ ) (i.e., φ ( T M ) ⊂ T M ⊥ ).Then we get m c + 34 ≤ △ ff ≤ m c + 34 . Proof.
By Lemma 2.1 and (3.14), we obtain the result. (cid:3)
Lemma 3.19.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Sasakian space form of constant φ -sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a totally geodesic φ -totally real submanifold ( M, g ) of ( M , g ) such that either the warping function f is not an eigen-function or the eigenvalueof f is not equal to m ( c +3)4 . Corollary 3.20.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product mani-fold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Sasakian space form of constant φ -sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume thatthe manifold ( M, g ) is a 2-dimensional totally geodesic submanifold of ( M , g ) with ξ ∈ Γ( T M ) .Then we get m · ≤ △ ff ≤ m · Lemma 3.21.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Sasakian space form of constant φ -sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a 2-dimensional totally geodesic submanifold ( M, g ) of ( M , g ) with ξ ∈ Γ( T M ) such that either the warping function f is not an eigen-function or the eigenvalue of f is not equal to m . KWANG-SOON PARK
Corollary 3.22.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product mani-fold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Sasakian space form of constant φ -sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume thatthe manifold ( M, g ) is a 2-dimensional totally geodesic φ -invariant submanifold of ( M , g ) with ξ ∈ Γ( T M ⊥ ) (i.e., φ ( T M ) =
T M ).Then we have m c ≤ △ ff ≤ m c Lemma 3.23.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Sasakian space form of constant φ -sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a 2-dimensional totally geodesic φ -invariant submanifold ( M, g ) of ( M , g ) with ξ ∈ Γ( T M ⊥ ) such that either the warping function f is notan eigen-function or the eigenvalue of f is not equal to m c . Theorem 3.24.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Kenmotsu space form of constant φ -sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Then we obtain m m m − H − m || h || − m ≤ △ ff ≤ m m H + m c, c ≥ − , (3.15) m m m − H − m || h || + m c ≤ △ ff ≤ m m H − m , c < − , (3.16) where m = dim B and m = dim F with m = m + m .Proof. We know that the Riemannian curvature tensor R [7] of ( M , g ) is given by R ( X, Y ) Z = c −
34 ( g ( Y, Z ) X − g ( X, Z ) Y )(3.17) + c + 14 ( η ( X ) η ( Z ) Y − η ( Y ) η ( Z ) X + η ( Y ) g ( X, Z ) ξ − η ( X ) g ( Y, Z ) ξ + g ( φY, Z ) φX − g ( φX, Z ) φY − g ( φX, Y ) φZ )for X, Y, Z ∈ Γ( T M ). Given orthonormal vectors
X, Y ∈ T p M , p ∈ M , we have(3.18) K ( X ∧ Y ) = R ( X, Y, Y, X ) = c −
34 + c + 14 ( − η ( Y ) − η ( X ) +3 g ( φX, Y ) ) . so that since − ≤ − η ( Y ) − η ( X ) + 3 g ( φX, Y ) ≤
3, we get − ≤ K ( X ∧ Y ) ≤ c, c ≥ − ,c ≤ K ( X ∧ Y ) ≤ − , c < − . From Lemma 2.1, we obtain the result. (cid:3)
Corollary 3.25.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product mani-fold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Kenmotsu space form of constant φ -sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume thatthe manifold ( M, g ) is a 2-dimensional totally geodesic submanifold of ( M , g ) with ξ ∈ Γ( T M ) .Then we get m · ( − ≤ △ ff ≤ m · ( − . N THE ESTIMATES OF WARPING FUNCTIONS 9
Lemma 3.26.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Kenmotsu space form of constant φ -sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a 2-dimensional totally geodesic submanifold ( M, g ) of ( M , g ) with ξ ∈ Γ( T M ) such that either the warping function f is not an eigen-function or the eigenvalue of f is not equal to − m . Corollary 3.27.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product mani-fold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Kenmotsu space form of constant φ -sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume thatthe manifold ( M, g ) is a 2-dimensional totally geodesic φ -invariant submanifold of ( M , g ) with ξ ∈ Γ( T M ⊥ ) (i.e., φ ( T M ) =
T M ).Then we get m c ≤ △ ff ≤ m c. Lemma 3.28.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Kenmotsu space form of constant φ -sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a 2-dimensional totally geodesic φ -invariant submanifold ( M, g ) of ( M , g ) with ξ ∈ Γ( T M ⊥ ) such that either the warping function f is notan eigen-function or the eigenvalue of f is not equal to m c . Corollary 3.29.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product mani-fold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Kenmotsu space form of constant φ -sectionalcurvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume thatthe manifold ( M, g ) is a totally geodesic φ -totally real submanifold of ( M , g ) with ξ ∈ Γ( T M ⊥ ) (i.e., φ ( T M ) ⊂ T M ⊥ ).Then we have m c − ≤ △ ff ≤ m c − . Lemma 3.30.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = ( M ( c ) , φ, ξ, η, g ) a Kenmotsu space form of constant φ -sectional curvature c . Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a totally geodesic φ -totally real submanifold ( M, g ) of ( M , g ) with ξ ∈ Γ( T M ⊥ ) such that either the warping function f is not an eigen-functionor the eigenvalue of f is not equal to m ( c − . Hermitian symmetric spaces
Theorem 4.1.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = G ( C m +2 ) = SU m +2 /S ( U m U ) the complex two-plane Grassmannian. Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Then we have (4.1) m m m − H − m || h || − m ≤ △ ff ≤ m m H + 8 m , where m = dim B and m = dim F with m = m + m . Proof.
We see that the Riemannian curvature tensor R [12] of ( M , g ) is given by R ( X, Y ) Z = g ( Y, Z ) X − g ( X, Z ) Y (4.2) + g ( JY, Z ) JX − g ( JX, Z ) JY − g ( JX, Y ) JZ + X α =1 ( g ( J α Y, Z ) J α X − g ( J α X, Z ) J α Y − g ( J α X, Y ) J α Z )+ X α =1 ( g ( J α JY, Z ) J α JX − g ( J α JX, Z ) J α JY )for X, Y, Z ∈ Γ( T M ). Given orthonormal vectors
X, Y ∈ T p M , p ∈ M , we get K ( X ∧ Y ) = R ( X, Y, Y, X ) = 1 + 3 g ( JX, Y ) (4.3) + X α =1 (3 g ( J α X, Y ) + g ( J α JY, Y ) g ( J α JX, X ) − g ( J α JX, Y ) ) . With simple computations, we obtain g ( JX, Y ) ≤ | JX | | Y | = 1 , X α =1 g ( J α X, Y ) ≤ | Y | = 1 (since { J X, J X, J X } is orthonormal), | X α =1 g ( J α JY, Y ) g ( J α JX, X ) | ≤ vuut X α =1 g ( J α JY, Y ) · vuut X α =1 g ( J α JX, X ) ≤ p | Y | p | X | = 1(by Cauchy-Schwarz inequality and since { J JY, J JY, J JY } and { J JX, J JX, J JX } are orthonormal) ⇒ − ≤ X α =1 g ( J α JY, Y ) g ( J α JX, X ) ≤ X α =1 g ( J α JX, Y ) ≤ | Y | = 1 (since { J JX, J JX, J JX } is orthonormal).By using the above relations, we obtain(4.4) K ( X ∧ Y ) ≤ · · . On the other hand, by the above relations, we have K ( X ∧ Y ) ≥ X α =1 ( g ( J α JY, Y ) g ( J α JX, X ) − g ( J α JX, Y ) )(4.5) ≥ − − − . From Lemma 2.1, by using (4.4) and (4.5), the result follows. (cid:3)
Remark . Choose orthonormal vectors
X, Y ∈ T p M , p ∈ M such that Y = JX and X is a singular vector. i.e., conveniently, JX = J X (See [1]). From (4.3), weget K ( X ∧ Y ) = 1 + 3 + 3 + 1 + 0 = 8 . So, the upper bound of the function K ( X ∧ Y ) is rigid. N THE ESTIMATES OF WARPING FUNCTIONS 11
Corollary 4.3.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = G ( C m +2 ) = SU m +2 /S ( U m U ) the complex two-plane Grassmannian. Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume that the manifold ( M, g ) isa 2-dimensional totally geodesic J -invariant submanifold of ( M , g ) with a singularvector field X ∈ Γ( T M ) (i.e., J ( T M ) =
T M ).Then we get m · ≤ △ ff ≤ m · . Proof.
By Lemma 2.1 and (4.3), we obtain the result. (cid:3)
Lemma 4.4.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = G ( C m +2 ) = SU m +2 /S ( U m U ) the complex two-plane Grassmannian.Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a 2-dimensional totally geodesic J -invariant submanifold ( M, g ) of ( M , g ) with a singular vector field X ∈ Γ( T M ) such that either the warp-ing function f is not an eigen-function or the eigenvalue of f is not equal to m . Theorem 4.5.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = SU ,m /S ( U · U m ) the complex hyperbolic two-plane Grassmannian.Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Then we obtain (4.6) m m m − H − m || h || − m ≤ △ ff ≤ m m H + 12 m , where m = dim B and m = dim F with m = m + m .Proof. We know that the Riemannian curvature tensor R [12] of ( M , g ) is given by R ( X, Y ) Z = −
12 ( g ( Y, Z ) X − g ( X, Z ) Y (4.7) + g ( JY, Z ) JX − g ( JX, Z ) JY − g ( JX, Y ) JZ + X α =1 ( g ( J α Y, Z ) J α X − g ( J α X, Z ) J α Y − g ( J α X, Y ) J α Z )+ X α =1 ( g ( J α JY, Z ) J α JX − g ( J α JX, Z ) J α JY ))for X, Y, Z ∈ Γ( T M ).Hence, in a similar way with Theorem 4.1, we easily get the result. (cid:3)
Remark . We choose orthonormal vectors
X, Y ∈ T p M , p ∈ M , such that Y = JX and X is a singular vector. i.e., conveniently, JX = J X (See [2]). In asimilar way with Remark 4.2, we obtain K ( X ∧ Y ) = − . So, the lower bound of the function K ( X ∧ Y ) is rigid. Corollary 4.7.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifoldand ( M , g ) = SU ,m /S ( U · U m ) the complex hyperbolic two-plane Grassmannian.Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume that the manifold ( M, g ) is a 2-dimensional totally geodesic J -invariant submanifold of ( M , g ) with asingular vector field X ∈ Γ( T M ) Then we get m · ( − ≤ △ ff ≤ m · ( − . Lemma 4.8.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = SU ,m /S ( U · U m ) the complex hyperbolic two-plane Grassmannian. Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a 2-dimensional totally geodesic J -invariant submanifold ( M, g ) of ( M , g ) with a singular vector field X ∈ Γ( T M ) such that either the warp-ing function f is not an eigen-function or the eigenvalue of f is not equal to − m . Theorem 4.9.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = Q m = SO m +2 /SO m SO the complex quadric. Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Then we get (4.8) m m m − H − m || h || − . m ≤ △ ff ≤ m m H + 5 m , where m = dim B and m = dim F with m = m + m .Proof. We see that the Riemannian curvature tensor R [13] of ( M , g ) is given by R ( X, Y ) Z = g ( Y, Z ) X − g ( X, Z ) Y (4.9) + g ( JY, Z ) JX − g ( JX, Z ) JY − g ( JX, Y ) JZ + g ( AY, Z ) AX − g ( AX, Z ) AY + g ( JAY, Z ) JAX − g ( JAX, Z ) JAY for
X, Y, Z ∈ Γ( T M ). Given orthonormal vectors
X, Y ∈ T p M , p ∈ M , we obtain K ( X ∧ Y ) = R ( X, Y, Y, X ) = 1 + 3 g ( JX, Y ) (4.10) + g ( AY, Y ) g ( AX, X ) − g ( AX, Y ) + g ( JAY, Y ) g ( JAX, X ) − g ( JAX, Y ) . Since A is an involution (i.e., A = id ), we get the following decompositions X = aX + bX Y = cY + dY , where | X | = | X | = | Y | = | Y | = 1, X , Y ∈ V ( A ) = { Z ∈ T p M | AZ = Z } , X , Y ∈ JV ( A ) (See [13]) so that1 = | X | = a + b , | Y | = c + d , g ( X, Y ) = acg ( X , Y ) + bdg ( X , Y ) . Conveniently, let ( a, b ) = (cos α, sin α ) and ( c, d ) = (cos β, sin β ) . If necessary, by replacing X , X , Y , Y with − X , − X , − Y , − Y , respec-tively, we may assume(4.11) 0 ≤ α, β ≤ π . Thus, with a simple calculation, we have K ( X ∧ Y ) = 1 + 2 a cos α sin β + 2 b sin α cos β + cos 2 α cos 2 β (4.12) +2 ab sin 2 α sin 2 β + cd sin 2 α sin 2 β − e cos α cos β, N THE ESTIMATES OF WARPING FUNCTIONS 13 where a = g ( X , JY ) b = g ( X , JY ) c = g ( JY , Y ) d = g ( JX , X ) e = g ( X , Y ) . We see(4.13) − ≤ a, b, c, d, e ≤ . Consider the function S ( x, y ) = 2 a cos x sin y + 2 b sin x cos y + cos 2 x cos 2 y (4.14) +2 ab sin 2 x sin 2 y + cd sin 2 x sin 2 y − e cos x cos y for ( x, y ) ∈ [0 , π ] × [0 , π ].Since sin 2 x sin 2 y ≥
0, by (4.13), we obtain S ( x, y ) ≤ x sin y + 2 sin x cos y (4.15) + cos 2 x cos 2 y + 2 sin 2 x sin 2 y + sin 2 x sin 2 y = 2(cos x sin y + sin x cos y ) + cos(2 x − y ) + sin 2 x sin 2 y = 2 sin ( x + y ) + cos(2 x − y ) + sin 2 x sin 2 y ≤ S ( x, y ) ≥ cos 2 x cos 2 y − x sin 2 y (4.16) − sin 2 x sin 2 y − cos x cos y = cos(2 x + 2 y ) − x sin 2 y − cos x cos y. Consider the function h ( x, y ) = cos(2 x + 2 y ) − x sin 2 y − cos x cos y for( x, y ) ∈ [0 , π ] × [0 , π ].We see(4.17) h ( x, y ) ≥ − . (cid:3) Remark . We get h ( π , π ) = − .
25. But h x ( π , π ) = = 0 and h y ( π , π ) = =0, which implies that ( π , π ) is not a critical point of h ( x, y ). Corollary 4.11.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product mani-fold and ( M , g ) = Q m = SO m +2 /SO m SO the complex quadric. Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume that the manifold ( M, g ) is a 2-dimensionaltotally geodesic J -invariant submanifold of ( M , g ) with a non-vanishing vector field X ∈ Γ( T M ) ∩ V ( A ) .Then we get m · ≤ △ ff ≤ m · . Proof.
By Lemma 2.1 and (4.10), we obtain the result. (cid:3) z = cos(2x+2y)–2sin(2x) sin(2y)-cos(x)^2 cos(y)^2, z = –3.200.20.40.60.811.21.41.6 x 00.20.40.60.811.21.4 y–3–2–101 (a) z = h ( x, y ) and z = − . z = cos(2x+2y)–2sin(2x) sin(2y)-cos(x)^2 cos(y)^2, z = –3.300.20.40.60.811.21.41.6 x 00.20.40.60.811.21.4 y–3–2–101 (b) z = h ( x, y ) and z = − . Figure 1.
The lower bound of h ( x, y ) Lemma 4.12.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = Q m = SO m +2 /SO m SO the complex quadric. Let ψ : ( M, g ) ( M , g ) be an isometric immersion. N THE ESTIMATES OF WARPING FUNCTIONS 15
There does not exist a 2-dimensional totally geodesic J -invariant submanifold ( M, g ) of ( M , g ) with a non-vanishing vector field X ∈ Γ( T M ) ∩ V ( A ) such thateither the warping function f is not an eigen-function or the eigenvalue of f is notequal to m . Corollary 4.13.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product mani-fold and ( M , g ) = Q m = SO m +2 /SO m SO the complex quadric. Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume that the manifold ( M, g ) is a 2-dimensionaltotally geodesic submanifold of ( M , g ) with T M ⊂ V ( A ) .Then we get m · ≤ △ ff ≤ m · . Lemma 4.14.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = Q m = SO m +2 /SO m SO the complex quadric. Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a 2-dimensional totally geodesic submanifold ( M, g ) of ( M , g ) with T M ⊂ V ( A ) such that either the warping function f is not an eigen-function or the eigenvalue of f is not equal to m . Corollary 4.15.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product mani-fold and ( M , g ) = Q m = SO m +2 /SO m SO the complex quadric. Let ψ : ( M, g ) ( M , g ) be an isometric immersion. Assume that the manifold ( M, g ) is a 2-dimensionaltotally geodesic submanifold of ( M , g ) with T M ⊥ J ( T M ) and dim( T M ∩ V ( A )) =dim( T M ∩ JV ( A )) = 1 .Then we get m · ≤ △ ff ≤ m · . Lemma 4.16.
Let ( M, g ) = ( B × f F, g B + f g F ) be a warped product manifold and ( M , g ) = Q m = SO m +2 /SO m SO the complex quadric. Let ψ : ( M, g ) ( M , g ) be an isometric immersion.There does not exist a 2-dimensional totally geodesic submanifold ( M, g ) of ( M , g ) with T M ⊥ J ( T M ) and dim( T M ∩ V ( A )) = dim( T M ∩ JV ( A )) = 1 suchthat the warping function f is not a harmonic function. Open questions
In section 3 and section 4, we deal with the estimates of the functions △ ff forisometric immersions ψ : ( M, g ) = ( B × f F, g B + f g F ) ( M , g ). And we alsoconsider equality cases and their applications. As future projects, we can use theseresults to study the properties of base manifolds and target manifolds and investi-gate another equality cases and their applications. We also estimate the functions △ ff by changing target manifolds. Questions
1. What kind of eigenvalues of the warping functions f can we get?(We obtained the following eigenvalues: m c, m c , m ( c + 3)4 , m , m ( c − , m , − m , m , f is an eigen-function with eigenvalue d , then whatcan we say about M and M ? References [1] J. Berndt, Y. J. Suh,
Real hypersurfaces in complex two-plane Grassmannians , Monatsh.Math. (1999), , 1-14.[2] J. Berndt, Y. J. Suh,
Hypersurfaces in noncompact complex Grassmannians of rank two ,Internat. J. Math. (2012), , 1250103(35 pages).[3] B. Y. Chen, On isometric minimal immersions from warped products into real space forms ,Proc. Edinburgh Math. Soc., (2002), , No. 3, 579-587.[4] B. Y. Chen, Warped products in real space forms , Rocky Mountain J. Math., (2004), , No.2, 551-563.[5] B. Y. Chen, Geometry of warped product and CR-warped product submanifolds in K¨ahlermanifolds: modified version , arXiv:1806.11102 [math.DG].[6] S. Ishihara,
Quaternion K¨ahlerian manifolds , J. Diff. Geometry, (1974), , 483-500.[7] K. Kenmotsu, A class of almost contact Riemannian manifolds , Tohoku Mathematical Jour-nal (1972), , No. 1, 93-103.[8] S. Kobayashi, K. Nomizu, Foundations of differential geometry , John Wiley & Sons, NewYork, 1996.[9] J. F. Nash,
The imbedding problem for Riemannian manifolds , Ann. of Math., (1956), ,20-63.[10] K. Ogiue, On almost contact manifolds admitting axiom of planes or axiom of free mobility ,Kodai Math. Sem. Rep., (1964), , No. 4, 223-232.[11] K. S. Park, Warped products in Riemannian manifolds , Bull. Aust. Math. Soc., (2014), ,510-520.[12] K. S. Park, Inequalities for the Casorati curvatures of real hypersurfaces in some Grassman-nians , Taiwan. J. Math., (2018), , No. 1, 63-77.[13] Y. J. Suh, Real hypersurfaces in the complex quadric with parallel Ricci tensor , Advances inMath., (2015), , 886-905.
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