Cirnu M., Normal anti-invariant submanifolds with closed conformal ...

Bulletin of the Transilvania University of Braşov • Vol 3(52) - 2010 

Series III: Mathematics, Informatics, Physics, 9-18 

NORMAL ANTI-INVARIANT SUBMANIFOLDS WITH 

CLOSEDCONFORMAL VECTOR FIELD IN KENMOTSU 

MANIFOLDS 

Maria CÎRNU 1 

Abstract 

We study some topological and geometric properties of normal anti-invariant submanifolds 

with closed conformal vector fields in Kenmotsu manifolds. 

2000 Mathematics Subject Classification: 53C42, 53C25, 53D15. 

Key words: Kenmotsu manifold, normal anti-invariant submanifold, special closed 

and conformal vector field, Whitney sphere. 

1 Indroduction and preliminaries 

Very interesting results about the study of Lagrangian submanifolds with closed conformal 

vector field in C n were obtained in [8]. Later, in [6], integral submanifolds with 

closed conformal vector field in Sasakian manifolds were analyzed and some topological 

and geometric properties of them were given. In this article we study similar properties 

of special closed and conformal vector field of normal anti-invariant submanifolds in Kenmotsu 

manifolds. We also study a class of these submanifolds in Kenmotsu space forms 

(called Whitney spheres), having a special closed and conformal vector field. 

Now, we review some notions and results from contact geometry, useful for our next 

considerations. 

Let ˜M be a C ∞ –differentiable, 2n + 1–dimensional almost contact manifold with the 

almost contact metric structure (F, ξ, η, g), where F is a (1, 1)–tensor field, η is a 1-form, g 

is a Riemannian metric on ˜M, ξ is the Reeb vector field, all these satisfying the following 

conditions : 

F 2 = −I + η ⊗ ξ; η(ξ) = 1; g(F X, F Y ) = g(X, Y ) − η(X)η(Y ) (1) 

for X, Y in χ(˜M), where χ(˜M) represents the set of all vector fields on ˜M. 

Let M be a submanifold of ˜M. We consider ∇ the Levi-Civita connection induced by 

˜∇ on M, ∇ ⊥ the connection in the normal bundle T ⊥ (M), h the second fundamental form 

1 Faculty of Mathematics and Informatics, Transilvania University of Braşov, Romania, e-mail: 

maria.cirnu@yahoo.com

10 Maria Cîrnu 

on M ¸and A ⃗n the Weingarten operator. The well-known Gauss–Weingarten formulas on 

M are: 

˜∇ X Y = ∇ X Y + h(X, Y ); ˜∇X ⃗n = −A ⃗n X + ∇ ⊥ X⃗n (2) 

for X, Y in χ(M) and ⃗n in χ ⊥ (M), where χ ⊥ (M) represents the set of normal vector fields 

on M. 

We also consider the fundamental 2-form Ω on ˜M, given by Ω(X, Y ) = g(X, F Y ) and 

denote by N F the Nijenhius tensor of F . It is known that ˜M is a Kenmotsu manifold if 

and only if 

dη = 0; dΩ = 2η ∧ Ω; N (1) = N F + 2dη ⊗ ξ = 0, 

or equivalently 

( ˜∇ X F )Y = g(Y, F X)ξ − η(Y )F X, (3) 

for X, Y in χ(˜M). 

A submanifold M, normal to the Reeb vector field ξ in a Kenmotsu manifold ˜M, is a 

normal semi-invariant submanifold, [7], if there are D, D ⊥ two orthogonal distributions 

on M so that: 

1. T x M = D x ⊕ D ⊥ x ; 

2. D is invariant, that is F D x = D x ; 

3. D ⊥ is anti-invariant, that is F D ⊥ x ⊆ T ⊥ x M, 

for all x in M, where T x M represents the tangent space in x on M and T ⊥ x M the normal 

space in x on M. Moreover, if D ⊥ = 0 then M is a normal invariant submanifold of ˜M, 

and if D = 0 then M is a normal anti-invariant submanifold of ˜M. 

An important property on these submanifolds, given in [7], and useful for our considerations 

is the following: 

Proposition 1. Let M be a normal semi-invariant submanifold of the Kenmotsu manifold 

˜M. Then: 

i) A F X Y = A F Y X, for all X, Y ∈ D ⊥ ; 

ii) A ξ Z = −Z and ∇ ⊥ Zξ = 0, for all Z ∈ χ(M). 

From [4], we have the following expression of the curvature tensor in Kenmotsu space 

forms: 

˜R(X, Y )Z = c − 3 [g(Y, Z)X − g(X, Z)Y ] + c + 1 

4 

4 [η(X)η(Z)Y 

− η(Y )η(Z)X + g(X, Z)η(Y )ξ − g(Y, Z)η(X)ξ + Ω(X, Z)F Y 

− Ω(Y, Z)F X + 2Ω(X, Y )F Z], (4) 

for all X, Y, Z ∈ χ(˜M), where ˜R is the curvature tensor of ˜M.

Normal anti-invariant submanifolds with closed conformal vector field 11 

2 Some properties of normal anti-invariant submanifolds of 

a Kenmotsu manifold 

Let M be an m-dimensional normal anti-invariant submanifold of 2n + 1-dimensional 

Kenmotsu manifold ˜M. Because ξ⊥M, F /TxM is one to one, for x ∈ M, we obtain that 

m ≤ n. We consider the following descomposition: 

T ⊥ x M = F T x M ⊕ τ x (M)⊕ < ξ x >, 

where τ x (M) is the complement of F T x M⊕ < ξ x > in T ⊥ x M. We also consider the bundles 

F T M = ⋃ x∈M F T xM, τ(M) = ⋃ x∈M τ x(M). Let {e 1 , ..., e m } be an orthonormal basis on 

M. Then {F e 1 , ..., F e m } is an orthonormal basis of F T M and we consider {e 2m+1 , ..., e 2n } 

an orthonormal basis in τ(M) and put e 2n+1 = ξ. 

For X, Y ∈ χ(M) we denote by h τ (X, Y ) and h F (X, Y ) the component of h(X, Y ) on 

τ(M) and on F T M, respectively. 

Using (2), (3) and Proposition 1. ii), we obtain the following proposition: 

Proposition 2. If M is m-dimensional then: 

∇ ⊥ ◦ F = F ◦ ∇ + F ◦ h τ ; F ◦ h F + A F = 0; h = h F + h τ − g ⊗ ξ. (5) 

Remark 1. If M is an n-dimensional normal anti-invariant submanifold of 2n + 1- dimensional 

Kenmotsu manifold ˜M, then τ(M) = 0, h τ = 0 and 

We define the 3-form 

∇ ⊥ ◦ F = F ◦ ∇; h = h F − g ⊗ ξ. (6) 

C F : χ(M) × χ(M) × χ(M) → F(M); C F (X, Y, Z) = g(h F (X, Y ), F Z), (7) 

for all X, Y, Z ∈ χ(M), where F(M) is the set of all real valued smooth functions on 

M. From the symmetry of h, Proposition 1 and (5), it results that C F is symmetric and 

3-linear. 

Proposition 3. Let M be an m-dimensional normal anti-invariant submanifold of 2n+1- 

dimensional Kenmotsu manifold ˜M. Then the 3-linear form ∇C F is symmetric in X and 

Y . 

Proof. By a straighforward computation and using the properties of Levi-Civita connection, 

Proposition 1, (2), (3), (5) and the definitions of ∇h F and ∇C F , we obtain: 

g((∇ X h F )(Y, Z) − (∇ Y h F )(X, Z), F U) 

= X(g(h F (Y, Z), F U)) − g(h F (∇ X Y, Z), F U) − g(h F (Y, ∇ X Z), F U) 

− g(h F (Y, Z), F ∇ X U) − Y (g(h F (X, Z), F U)) + g(h F (∇ Y X, Z), F U) 

+ g(h F (X, ∇ Y Z), F U) + g(h F (X, Z), F ∇ Y U) 

= (∇ X C F )(Y, Z, U) − (∇ Y C F )(X, Z, U), (8)


for all X, Y, Z, U ∈ χ(M). Moreover, because M is a normal anti-invariant submanifold, 

from (4), we have: 

[ ˜R(X, Y )Z] ⊥ = 0, 

for all X, Y, Z ∈ χ(M). Now, from the Codazzi equation and (8), it results that ∇C F is 

symmetric in X and Y . 

For ⃗n ∈ χ ⊥ (M), we consider the 1-form α ⃗n : χ(M) → F (M), given by 

for all X ∈ χ(M). We have the following proposition: 

α ⃗n (X) = g(F⃗n, X), (9) 

Proposition 4. Let M be an m-dimensional normal anti-invariant submanifold of 2n+1- 

dimensional Kenmotsu manifold ˜M. Then: 

i) α ⃗n is closed if and only if 

for all X, Y ∈ χ(M), 

ii) α ⃗n = 0, for all ⃗n ∈ τ(M)⊕ < ξ >. 

g(∇ ⊥ Y ⃗n, F X) = g(∇ ⊥ X⃗n, F Y ), (10) 

Proof. i) From the properties of Levi-Civita connection and (3), it results: 

(dα ⃗n )(X, Y ) = −g(∇ ⊥ X⃗n, F Y ) + g(∇ ⊥ Y ⃗n, F X) 

and then (10). 

ii) results from (9) and the descomposition of χ ⊥ (M) = F T M ⊕ τ(M)⊕ < ξ > . 

A vector field X on a normal anti-invariant submanifold M of a Kenmotsu manifold 

˜M is closed if α F X is a closed 1-form. 

Proposition 5. Let M be an m-dimensional normal anti-invariant submanifold of a 2n+ 

1-dimensional Kenmotsu manifold ˜M. If one of the following conditions is satisfied: 

i) M has parallel mean curvature vector ; 

ii) m = n, 

then F H is a closed vector field. 

Proof. i) Let H be the mean curvature vector of M. Because M is parallel, we have (10), 

that is α H is a closed form. Then α F 2 H = α −H+η(H)ξ = −α H is closed. 

ii) m = n implies H = H F and α F H (X) = g(F 2 H, X) = g(−H + η(H)ξ, X) = 0, for 

X ∈ χ(M). 

Definition 1. Let M be a normal anti-invariant submanifold of a Kenmotsu manifold ˜M. 

A vector field X ∈ χ(M) is a closed conformal vector field if 

i) X is closed; 

ii) X is conformal, that is 

g(∇ Y X, Z) + g(∇ Z X, Y ) = 

2 

div(X), (11) 

dim M


for all Y, Z ∈ χ(M), where div(X) represents the divergence of X. 

Moreover, if there is f ∈ F (M) such as h(X, X) = fF X, then X is a special closed 

conformal vector field on M. 

Remark 2. Conditions i) and ii) from Definition 1 are equivalent to: 

for all Y ∈ χ(M). 

We consider 

∇ Y X = div(X) Y, (12) 

dim M 

L = { Y/Y ∈ χ(M), g(F H F , Y ) = 0 } . (13) 

Proposition 6. Let M be a m-dimensional normal anti-invariant submanifold of a 2n+1- 

dimensional Kenmotsu manifold ˜M and X a closed special conformal vector field on M. 

Then: 

i) X is an eigenvector of A F X , corresponding to the eigenvalue f; 

ii) Y is an eigenvector of A F X , corresponding to the eigenvalue λ if and only if 

h F (X, Y ) = λF X; 

∇ ⊥ Y (F X) = div(X) 

m F Y + F hτ (X, Y ); (14) 

iii) for all Y, Z ∈ L eigenvectors, corresponding to different eigenvalues of A F X , Y, Z 

are orthogonal and C F (X, Y, Z) = 0. 

Proof. i) From (5), A F X X = −F h F (X, X) = −F (fX) = −fF 2 X = fX. 

ii) Let Y be a an eigenvector corresponding to A F X with λ its eigenvalue. From (5), 

A F X Y = −F h F (X, Y ) = λY and then h F (X, Y ) = λF Y . Because h F is symmetric, we 

have h F (Y, X) = λF X and then h F (X, Y ) = λF X. 

The second equality results from (2), (3) and (12). 

iii) Let λ 1 be the eigenvalue of Y and λ 2 the eigenvalue of Z, so that λ 1 ≠ λ 2 . We 

have C F (Y, Z, X) = g(h(Y, Z), F X) = g(A F X Y, Z) = λ 1 g(Y, Z) and C F (Z, Y, X) = 

g(h(Z, Y ), F X) = g(A F X Z, Y ) = λ 2 g(Z, Y ). The affirmation results from the fact that 

C F is symmetric and λ 1 ≠ λ 2 . 

Now, we recall from [8] two lemmas, useful for our next considerations: 

Lemma 1. Let X be a nontrivial, closed and conformal vector field on an n-dimensional 

Riemannian manifold M with the metric g. Then: 

i) The curvature tensor R of M satisfies the equation: 

‖X‖ 2 R(V, W )X = Ric(X) [g(W, X) − g(V, X)W ]; (15) 

n − 1 

ii) IF M ′ = {p ∈ M/X p ≠ 0}, then the distribution 

defines an umbilical foliation on M. 

D : p ∈ M ′ → D(p) = {V ∈ T p M/g(V, X) = 0}


Lemma 2. Let ψ : M → C n be a Lagrangian immersion of an n- dimensional manifold M 

and X a closed and conformal vector field without zeroes on M so that h(X, X) = fF X, 

where f is a differential function on M and F is the hermitian structure on M. Then: 

i) For any point p ∈ M, there are at most two eigenvalues of A F X on D(p) which 

satisfy the quadric equation: 

λ 2 − fλ + Ric(X) 

n − 1 

= 0, (16) 

where Ric represents the Ricci tensor of M. Moreover, the eigenvalues of A F X are constant 

on the connected leaves of the foliation D. 

ii) IF V, W ∈ D(p) are eigenvectors of A F X , corresponding to different eigenvalues, 

then h(V, W ) = 0. 

Proposition 7. Let M be an m-dimensional anti-invariant submanifold of the 2n + 1- 

dimensional Kenmotsu space form ˜M(c). We consider X a closed special conformal vector 

field on M and L X = {Y ∈ χ(M)/g(X, Y ) = 0}. If Y ∈ L X , Y ≠ 0, is an eigenvector of 

A F X , corresponding to eigenvalue λ, then: 

λ 2 − fλ + Ric(X) 

m − 1 − c − 3 ‖X‖ 2 + 1 

4 ‖Y ‖ 2 ‖hτ (X, Y )‖ 2 = 0. (17) 

Proof. From the Gauss equation, we obtain: 

g( ˜R(X, Y )X, Y ) = g(R(X, Y )X, Y ) − g(h(X, Y ), h(X, Y )) + g(h(X, X), h(Y, Y )), (18) 

where R is the curvature tensor of M. Because X is special, Y ∈ L X , using the descomposion 

(5) of h(X, Y ), it results that 

g(h(Y, Y ), h(X, X)) = fλ ‖Y ‖ 2 ; g(h(X, Y ), h(X, Y )) = λ 2 ‖Y ‖ 2 + ‖h τ (X, Y )‖ 2 . 

From (15), we have R(X, Y )X = − Ric(X) 

m−1 Y and from (4), ˜R(X, Y )X = − 

c−3 

4 ‖X‖2 Y . 

Using these last relations in (18), we obtain the desired result. 

From Proposition 7 and Lemma 2, it results: 

Proposition 8. If M is an n-dimensional normal anti-invariant submanifold of 2n + 1- 

dimensional Kenmotsu space form ˜M(c) and X is a special closed conformal vector field 

on M, then: 

i) A F X has at most three eigenvalues f, λ 1 , λ 2 in any point of M, where λ 1 , λ 2 verify 

the equation: 

λ 2 − fλ + Ric(X) 

n − 1 − c − 3 ‖X‖ 2 = 0; 

4 

ii) the eigenvalues λ 1 and λ 2 of A F X are constant on the connected leaves of the 

foliation L.


Proposition 9. Let M be a normal anti-invariant submanifold in Kenmotsu space form 

˜M(c) with dim M ≥ 2. If X is a nontrivial parallel vector field on M such as h(X, X) = 

fF X, then f = constant. 

Proof. Because X is special, using (3), the properties of Levi-Civita connection, the definition 

of ∇C F and the fact that X is parallel, for Y, Z ∈ χ(M), we obtain: 

g( ˜∇ Y (h(X, X)), F Z) = Y (f)g(X, Z); (∇ Y C F )(X, X, Z) = Y (f)g(X, Z). 

Now, from the symmetry of ∇C F , we have X(f)g(Y, Z) = Y (f)g(X, Z) and then f = 

constant. 

3 Whitney sphere of a Kenmotsu manifold 

Definition 2. An n-dimensional normal anti-invariant submanifold M of the 2n + 1- 

dimensional Kenmotsu manifold ˜M is a Whitney sphere if its second fundamental form h 

satisfies the equation: 

h(V, V ) = λ[g(V, V )H + 2g(H, F V )F V + 2 g(H, ξ)g(V.V )ξ], (19) 

n 

where H is the mean curvature vector field of M, V ∈ χ(M) and λ ∈ F (M). 

Remark 3. If M is a Whitney sphere of a Kenmotsu manifold, then λ = 

n 

n+2 . From 

Proposition 1.ii), g(H, ξ) = −1 and using the bilinearity and symmetry of h, we have: 

h(V, W ) = λ[g(V, W )H + g(H, F V )F W + g(H, F W )F V − 2 g(V, W )ξ], (20) 

n 

for all V, W ∈ χ(M). 

Theorem 1. If M is a Whitney sphere of Kenmotsu space form ˜M(c), then F H is a 

closed conformal vector field on M. 

Proof. Let Z, V be two vector fields M, so that V is an unitary vector field and Z⊥V . 

Using (4), we obtain ˜R(Z, V )V = c−3 

4 

and then [ ˜R(Z, V )V ] ⊥ = 0. From Codazzi equation, 

we obtain (∇ ⊥ Z h)(V, V ) = (∇⊥ V 

h)(Z, V ). By a straightforward computation, using the 

properties of Levi-Civita connection, (3), (5) and (20) we have: 

and then 

(∇ ⊥ Zh)(V, V ) = λ[∇ ⊥ ZH + 2g(∇ ⊥ ZH, F V )F V ]; 

(∇ ⊥ V h)(Z, V ) = λ[g(∇ ⊥ V H, F V )F Z + g(∇ ⊥ V H, F Z)F V ] 

∇ ⊥ ZH + 2g(∇ ⊥ ZH, F V )F V = g(∇ ⊥ V H, F V )F Z + g(∇ ⊥ V H, F Z)F V. (21) 

From this last relation, it results g(∇ ⊥ Z H, F V ) + 2g(F V, ∇⊥ Z H) = g(∇⊥ V 

H, F Z). Because 

M is a Whitney sphere, from Proposition 5, F H is closed and from (10) we have 

g(F V, ∇ ⊥ Z H) = g(F Z, ∇⊥ V H) and then g(F V, ∇⊥ ZH) = 0. Moreover, using (21), we obtain: 

(∇ Z F H) = F ∇ ⊥ ZH = −g(F V, ∇ ⊥ V H)Z 

and from (12) we obtain that F H is a conformal vector field.


Proposition 10. If M is a Whitney sphere of 2n + 1-dimensional Kenmotsu space form 

˜M(c), with H its mean curvature tensor field, then the Ricci tensor, scalar curvature and 

sectional curvature of M have the following expresions: 

i) 

ii) 

(n − 1)(c − 3) 

Ric(V ) = [ 

4 

+ 

+ 3n2 − 4 

] ‖V ‖2 

(n + 2) 2 

n 2 

(n + 2) 2 [n ‖V ‖2 ‖H‖ 2 + (n − 2)g 2 (F V, H)]; (22) 

ρ = 

n(n − 1)(c − 3) 

4 

+ 

2n(n − 1) 

(n + 2) 

+ n2 (n − 1) 

(n + 2) 

‖H‖ 2 ; (23) 

iii) 

K(X, Y ) = c − 3 

4 

+ 

where (X, Y ) is a 2-plane of M. 

n 2 

(n + 2) 2 [‖H‖2 + 3g 2 (H, F X) + 3g 2 (H, F Y ) + 4 n + 4 ], (24) 

n2 Proof. i) Let {e i } i=1,n 

be an orthonormal basis on M. From the Gauss equation, we have: 

Ric(V ) = 

n∑ 

g( ˜R(e 

n∑ 

n∑ 

i , V )V, e i ) + g(h(e i , e i ), h(V, V )) − g(h(e i , V ), h(e i , V )). 

i=1 

i=1 

i=1 

From (4) and (20), we obtain: 

n∑ 

g( ˜R(e i , V )V, e i ) = 

i=1 

(c − 3)(n − 1) 

4 

‖V ‖ 2 ; 

n∑ 

g(h(e i , e i ), h(V, V ))) = nλ[‖V ‖ 2 ‖H‖ 2 + 2g 2 (H, F V ) + 2 n ‖V ‖2 ]; 

i=1 

n∑ 

g(h(e i , V ), h(e i , V )) = 

i=1 

n 2 

(n + 2) 2 [2 ‖H‖2 ‖V ‖ 2 + 6g 2 (F V, H) 

+ ( 4 n + 4 n 2 − 1) ‖V ‖2 + ng 2 (H, F V )]. 

From these relations, it results (22). 

ii) From i), we have: 

Ric(e i ) = 

(n − 1)(c − 3) 

4 

+ 3n2 − 4 

(n + 2) 2 + n 3 

(n + 2) 2 ‖H‖2 + n2 (n − 2) 

(n + 2) 2 g(H, F e i) 

and then (23). 

iii) results from an analoguos calculus as that used in i).


Using the Cauchy identity, from Proposition 10, it results: 

Proposition 11. Let M be an Whitney sphere of a 2n + 1-dimensional Kenmotsu space 

form ˜M(c) with H its mean curvature vector field. Then, the Ricci tensor and the sectional 

curvature of M verify the following inequalities: 

i) 

≤ 

(n − 1)(c − 3) 

4 

(n − 1)(c − 3) 

4 

+ 3n2 − 4 

(n + 2) 2 + n 3 

(n + 2) 2 ≤ Ric 

+ 3n2 − 4 

(n + 2) 2 + 2n2 (n − 1) 

‖H‖ 2 : 

(n + 2) 

ii) 

K(X, Y ) ≥ c − 3 

4 

where {X, Y } is a 2-plane of M. 

+ 4n2 

(n + 2) 2 (‖H‖2 + 1 n + 1 n 2 ), 

An important clasical result about the existence of harmonic forms is in the following 

Bochner’s theorem: 

Theorem 2. On an orientable compact Riemannian manifold with the Ricci tensor positively 

defined there is no nonzero 1-harmonic forms. 

Proposition 12. Let ˜M(c) be a Kenmotsu space form. Then: 

i) there are not compact Whitney spheres with Ric > 0 and c < 3; 

ii) there are not compact Whitney spheres with c > 3. 

Proof. i) Suppose that there is a compact Whitney sphere M of ˜M(c) with Ric < 0 and 

c < 3. From Theorem 2, the first de Rham cohomolgy group of M, H 1 (M, R) = 0. 

Because M is a Whitney sphere, from Proposition 5, F H is closed, hence it is the gradient 

of a differential function f ∈ F (M). But M compact implies that M has critical points 

which are zeroes of H. Moreover, if {e i } i=1,n 

is a local orthonormal basis on M, using the 

Gauss equation, we have: 

ng(h(V, V ), H) = Ric(V ) + 

n∑ 

‖h(V, e i )‖ 2 − 

i=1 

(c − 3)(n − 1) 

4 

‖V ‖ 2 

and then ‖H‖ > 0. 

ii) From Proposition 11. i), we have Ric > 0 and we have a similar argument as that used 

in the proof of i).


References 

[1] Blair, D. E., Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 

509 (1976), Springer-Verlag. 

[2] Blair, D. E., Riemannian geometry of contact and simplectic manifolds. Birkhäuser, 

Boston, 203(2002). 

[3] Blair, D. E., Oubina, J. A., Conformal and related changes of metrics on the product 

of two almost contact metric manifolds. Publ. Mat. Barcelona, 34(1990), 199–207. 

Chen, B.Y., Geometry of Submanifolds, M. Dekker, New York, 1973. 

[4] Janssens, D., Vanhecke, L., Almost contact structures and curvature tensors. Kodāi 

Math. J., 4(1981), 1–27. 

[5] Kenmotsu, K., A class of almost contact Riemannian manifolds, Tohoku Math. J., 

24(1976), 93–103. 

[6] Pitiş. Gh., Integral submanifolds with closed conformal vector field in Sasakian manifolds, 

New York J. Math., 11(2005), 157–170. 

[7] Papaghiuc, N., Semi-invariant submanifolds in a Kenmotsu manifold. Rend. di Mat., 

4(1983), 607–622. 

[8] Ros, A., Urbano. F., Lagrangian submanifolds of C n with conformal Maslov form and 

Whitney sphere, J. Math. Soc. Japan 50(1998), 203–226. 

[9] Udrişte, C., On conformal vector fields, Tensor N. S., 46(1987), 265–270.

Cirnu M., Normal anti-invariant submanifolds with closed conformal ...

Create successful ePaper yourself

Delete template?

Save as template?