nach oben

Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2014 | Research

Inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms

verfasst von: Vladimir Slesar, Bayram Şahin, Gabriel-Eduard Vîlcu

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2014

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Patentsuche

Aus

Abstract

In this paper we prove two sharp inequalities that relate the normalized scalar curvature with the Casorati curvature for a slant submanifold in a quaternionic space form. Moreover, we show that in both cases, the equality at all points characterizes the invariantly quasi-umbilical submanifolds.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

1 Introduction

In a seminal paper published in the early 1990s, Chen [1] established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature, both being intrinsic invariants, and squared mean curvature, the main extrinsic invariant, initiating the theory of δ-invariants or the so-called Chen invariants; this turned out to be one of the most interesting modern research topic in differential geometry of submanifolds. These inequalities were further extended to many classes of submanifolds in different ambient spaces (for an extensive and comprehensive survey on this topic see [2]). For example, in a quaternionic Kähler setting, Chen-like inequalities were proved in [3‐11] and a set of open problems in the field was proposed recently in [12]. Moreover, new optimal inequalities involving δ-invariants were recently proved in [13‐20]. We also note that some interesting inequalities for the length of the second fundamental form of the warped product submanifolds were obtained recently in [21‐25].

On the other hand, the Casorati curvature of a submanifold in a Riemannian manifold is an extrinsic invariant defined as the normalized square of the length of the second fundamental form. It is well known that this notion extends the concept of the principal direction of a hypersurface of a Riemannian manifold to submanifolds of a Riemannian manifold and it was preferred by Casorati over the traditional Gauss curvature because corresponds better with the common intuition of curvature [26] (see also [27‐30] for the geometrical meaning and the importance of the Casorati curvatures). Therefore it is of great interest to obtain optimal inequalities for the Casorati curvatures of submanifolds in different ambient spaces. We note that in [31], Decu, Haesen and Verstraelen obtained some optimal inequalities involving the scalar curvature and the Casorati curvature of a Riemannian submanifold in a real space form and the holomorphic sectional curvature and the Casorati curvature of a Kähler hypersurface in a complex space form. Moreover, the same authors proved in [32] an inequality in which the scalar curvature is estimated from above by the normalized Casorati curvatures, while Ghişoiu obtained in [33] some inequalities for the Casorati curvatures of slant submanifolds in complex space forms.

In this paper we generalize these inequalities in a quaternionic setting, proving the following result which also solves the Problem 6.7 from [12].

Theorem 1.1 Let

M^{n}

be a θ-slant proper submanifold of a quaternionic space form

{\bar{M}}^{4 m} (c)

. Then:

(i)

The normalized δ-Casorati curvature

δ_{c} (n - 1)

satisfies

ρ \leq δ_{c} (n - 1) + \frac{c}{4} (1 + \frac{9}{n - 1} {cos}^{2} θ) .

(1)

Moreover, the equality sign holds if and only if

M^{n}

is an invariantly quasi-umbilical submanifold with trivial normal connection in

{\bar{M}}^{4 m} (c)

, such that with respect to suitable orthonormal tangent frame

{ξ_{1}, \dots, ξ_{n}}

and normal orthonormal frame

{ξ_{n + 1}, \dots, ξ_{4 m}}

, the shape operators

A_{r} \equiv A_{ξ_{r}}

r \in {n + 1, \dots, 4 m}

, take the following forms:

A_{n + 1} = (\begin{array}{cccccc} a & 0 & 0 & \dots & 0 & 0 \\ 0 & a & 0 & \dots & 0 & 0 \\ 0 & 0 & a & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & a & 0 \\ 0 & 0 & 0 & \dots & 0 & 2 a \end{array}), A_{n + 2} = \dots = A_{4 m} = 0 .

(2)

(ii)

The normalized δ-Casorati curvature

{\hat{δ}}_{c} (n - 1)

satisfies

ρ \leq {\hat{δ}}_{c} (n - 1) + \frac{c}{4} (1 + \frac{9}{n - 1} {cos}^{2} θ) .

(3)

Moreover, the equality sign holds if and only if

M^{n}

is an invariantly quasi-umbilical submanifold with trivial normal connection in

{\bar{M}}^{4 m} (c)

, such that with respect to suitable orthonormal tangent frame

{ξ_{1}, \dots, ξ_{n}}

and normal orthonormal frame

{ξ_{n + 1}, \dots, ξ_{4 m}}

, the shape operators

A_{r} \equiv A_{ξ_{r}}

r \in {n + 1, \dots, 4 m}

, take the following forms:

A_{n + 1} = (\begin{array}{cccccc} 2 a & 0 & 0 & \dots & 0 & 0 \\ 0 & 2 a & 0 & \dots & 0 & 0 \\ 0 & 0 & 2 a & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & 2 a & 0 \\ 0 & 0 & 0 & \dots & 0 & a \end{array}), A_{n + 2} = \dots = A_{4 m} = 0 .

(4)

2 Preliminaries

2.1 Riemannian invariants

In this subsection we recall some basic concepts in Riemannian geometry, using mainly [34].

Let

(\bar{M}, \bar{g})

be an m-dimensional Riemannian manifold. For an n-dimensional Riemannian submanifold M of

(\bar{M}, \bar{g})

, we denote by g the metric tensor induced on M. If

\bar{\nabla}

is the Levi-Civita connection on

\bar{M}

and ∇ is the covariant differentiation induced on M, then the Gauss and Weingarten formulas are given by

{\bar{\nabla}}_{X} Y = \nabla_{X} Y + h (X, Y), \forall X, Y \in Γ (T M)

and

{\bar{\nabla}}_{X} N = - A_{N} X + \nabla_{X}^{⊥} N, \forall X \in Γ (T M), \forall N \in Γ (T M^{⊥}),

where h is the second fundamental form of M,

\nabla^{⊥}

is the connection on the normal bundle and

A_{N}

is the shape operator of M with respect to N. The shape operator

A_{N}

is related to h by

g (A_{N} X, Y) = \bar{g} (h (X, Y), N)

for all

X, Y \in Γ (T M)

and

N \in Γ (T M^{⊥})

If we denote by

\bar{R}

and R the curvature tensor fields of

\bar{\nabla}

and ∇, then we have the Gauss equation:

\begin{array}{rcl} \bar{R} (X, Y, Z, W) & = & R (X, Y, Z, W) + \bar{g} (h (X, W), h (Y, Z)) \\ - \bar{g} (h (X, Z), h (Y, W)) \end{array}

(5)

for all

X, Y, Z, W \in Γ (T M)

We denote by

K (π)

the sectional curvature of M associated with a plane section

π \subset T_{p} M

p \in M

. If

{e_{1}, \dots, e_{n}}

is an orthonormal basis of the tangent space

T_{p} M

and

{e_{n + 1}, \dots, e_{m}}

is an orthonormal basis of the normal space

T_{p}^{⊥} M

, then the scalar curvature τ at p is given by

τ (p) = \sum_{1 \leq i < j \leq n} K (e_{i} \land e_{j})

and the normalized scalar curvature ρ of M is defined as

ρ = \frac{2 τ}{n (n - 1)} .

We denote by H the mean curvature vector, that is,

H (p) = \frac{1}{n} \sum_{i = 1}^{n} h (e_{i}, e_{i})

and we also set

h_{i j}^{α} = g (h (e_{i}, e_{j}), e_{α}), i, j \in {1, \dots, n}, α \in {n + 1, \dots, m} .

Then the squared mean curvature of the submanifold M in

\bar{M}

is defined by

{∥ H ∥}^{2} = \frac{1}{n^{2}} \sum_{α = n + 1}^{m} {(\sum_{i = 1}^{n} h_{i i}^{α})}^{2}

and the squared norm of h over dimension n is denoted by

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2014-123/MediaObjects/13660_2013_Article_1329_IEq26_HTML.gif

and is called the Casorati curvature of the submanifold M. Therefore we have

C = \frac{1}{n} \sum_{α = n + 1}^{m} \sum_{i, j = 1}^{n} {(h_{i j}^{α})}^{2} .

The submanifold M is called totally geodesic if the second fundamental form vanishes identically and totally umbilical if there is a real number λ such that

h (X, Y) = λ g (X, Y) H

for any tangent vectors X, Y on M. If

H = 0

, then the submanifold M is said to be minimal.

The submanifold M is called invariantly quasi-umbilical if there exist

m - n

mutually orthogonal unit normal vectors

ξ_{n + 1}, \dots, ξ_{m}

such that the shape operators with respect to all directions

ξ_{α}

have an eigenvalue of multiplicity

n - 1

and that for each

ξ_{α}

the distinguished eigendirection is the same [35].

Suppose now that L is an r-dimensional subspace of

T_{p} M

r \geq 2

and let

{e_{1}, \dots, e_{r}}

be an orthonormal basis of L. Then the scalar curvature

τ (L)

of the r-plane section L is given by

τ (L) = \sum_{1 \leq α < β \leq r} K (e_{α} \land e_{β})

and the Casorati curvature

C (L)

of the subspace L is defined as

C (L) = \frac{1}{r} \sum_{α = n + 1}^{m} \sum_{i, j = 1}^{r} {(h_{i j}^{α})}^{2} .

The normalized δ-Casorati curvature

δ_{c} (n - 1)

and

{\hat{δ}}_{c} (n - 1)

are given by

{[δ_{c} (n - 1)]}_{p} = \frac{1}{2} C_{p} + \frac{n + 1}{2 n (n - 1)} inf {C (L) | L a hyperplane of T_{p} M}

and

{[{\hat{δ}}_{c} (n - 1)]}_{p} = 2 C_{p} - \frac{2 n - 1}{2 n} sup {C (L) | L a hyperplane of T_{p} M} .

2.2 Quaternionic Kähler manifolds

We give in this subsection a quick review of basic definitions and properties of manifolds endowed with quaternionic Kähler structures, using mainly [36].

Let

\bar{M}

be a differentiable manifold and assume that there is a rank 3-subbundle σ of

End (T \bar{M})

such that a local basis

{J_{1}, J_{2}, J_{3}}

exists on sections of σ satisfying for all

α \in {1, 2, 3}

J_{α}^{2} = - Id, J_{α} J_{α + 1} = - J_{α + 1} J_{α} = J_{α + 2},

where Id denotes the identity tensor field of type

(1, 1)

on M and the indices are taken from

{1, 2, 3}

modulo 3. Then the bundle σ is called an almost quaternionic structure on M and

{J_{1}, J_{2}, J_{3}}

is called a canonical local basis of σ. Moreover,

(\bar{M}, σ)

is said to be an almost quaternionic manifold. It is easy to see that any almost quaternionic manifold is of dimension 4m,

m \geq 1

A Riemannian metric

\bar{g}

\bar{M}

is said to be adapted to the almost quaternionic structure σ if it satisfies

\bar{g} (J_{α} X, J_{α} Y) = \bar{g} (X, Y), \forall α \in {1, 2, 3}

for all vector fields X, Y on

\bar{M}

and any canonical local basis

{J_{1}, J_{2}, J_{3}}

of σ. Moreover,

(\bar{M}, σ, \bar{g})

is said to be an almost quaternionic Hermitian manifold.

If the bundle σ is parallel with respect to the Levi-Civita connection

\bar{\nabla}

\bar{g}

, then

(\bar{M}, σ, \bar{g})

is said to be a quaternionic Kähler manifold. Equivalently, locally defined 1-forms

ω_{1}

ω_{2}

ω_{3}

exist such that we have for all

α \in {1, 2, 3}

{\bar{\nabla}}_{X} J_{α} = ω_{α + 2} (X) J_{α + 1} - ω_{α + 1} (X) J_{α + 2}

for any vector field X on

\bar{M}

, where the indices are taken from

{1, 2, 3}

modulo 3.

We remark that any quaternionic Kähler manifold is an Einstein manifold, provided that

dim M > 4

Let

(\bar{M}, σ, \bar{g})

be a quaternionic Kähler manifold and let X be a non-null vector field on

\bar{M}

. Then the 4-plane spanned by

{X, J_{1} X, J_{2} X, J_{3} X}

, denoted by

Q (X)

, is called a quaternionic 4-plane. Any 2-plane in

Q (X)

is called a quaternionic plane. The sectional curvature of a quaternionic plane is called a quaternionic sectional curvature. A quaternionic Kähler manifold is a quaternionic space form if its quaternionic sectional curvatures are equal to a constant, say c. It is well known that a quaternionic Kähler manifold

(\bar{M}, σ, \bar{g})

is a quaternionic space form, denoted

\bar{M} (c)

, if and only if its curvature tensor is given by

\begin{array}{rcl} \bar{R} (X, Y) Z & = & \frac{c}{4} {\bar{g} (Z, Y) X - \bar{g} (X, Z) Y + \sum_{α = 1}^{3} [\bar{g} (Z, J_{α} Y) J_{α} X - \\ - \bar{g} (Z, J_{α} X) J_{α} Y + 2 \bar{g} (X, J_{α} Y) J_{α} Z]} \end{array}

(6)

for all vector fields X, Y, Z on

\bar{M}

and any local basis

{J_{1}, J_{2}, J_{3}}

of σ.

A submanifold M in a quaternionic Kähler manifold

(\bar{M}, σ, \bar{g})

is called a quaternionic submanifold [37] (resp. a totally real submanifold [38]) if each tangent space of M is carried into itself (resp. into the normal space) by each section in σ. In [39], the author introduced the concept of slant submanifolds as a natural generalization of both quaternionic and totally real submanifolds. A submanifold M of a quaternionic Kähler manifold

(\bar{M}, σ, \bar{g})

is said to be a slant submanifold if for each non-zero vector X tangent to M at p, the angle

θ (X)

between

J_{α} (X)

and

T_{p} M

α \in {1, 2, 3}

is constant, i.e. it does not depend on the choice of

p \in M

and

X \in T_{p} M

. We can easily see that quaternionic submanifolds are slant submanifolds with

θ = 0

and totally real submanifolds are slant submanifolds with

θ = \frac{π}{2}

. A slant submanifold of a quaternionic Käler manifold is said to be proper (or θ-slant proper) if it is neither quaternionic nor totally real. We note that another natural generalization of both quaternionic and totally real submanifolds in a quaternionic Kähler manifold is given by quaternionic CR-submanifolds. A submanifold M of a quaternion Kähler manifold

(\bar{M}, σ, \bar{g})

is said to be a quaternionic CR-submanifold if there exist two orthogonal complementary distributions D and

D^{⊥}

on M such that D is invariant under quaternionic structure and

D^{⊥}

is totally real (see [40]). It is clear that, although quaternionic CR-submanifolds are also the generalization of both quaternionic and totally real submanifolds, there exists no inclusion between the two classes of quaternionic CR-submanifolds and slant submanifolds.

We also note that we have the next characterization of slant submanifolds in quaternionic Kähler manifolds.

Theorem 2.1 [39]Let M be a submanifold of a quaternionic Kähler manifold

\bar{M}

. Then M is slant if and only if there exists a constant

λ \in [- 1, 0]

such that

P_{β} P_{α} X = λ X, \forall X \in Γ (T M), α, β \in {1, 2, 3},

(7)

where

P_{α} X

denote the tangential component of

J_{α} X

. Furthermore, in such a case, if θ is the slant angle of M, then it satisfies

λ = - {cos}^{2} θ

From the above theorem it follows easily that

g (P_{α} X, P_{β} Y) = {cos}^{2} θ g (X, Y)

(8)

for

X, Y \in Γ (T M)

and

α, β \in {1, 2, 3}

Moreover, every proper slant submanifold of a quaternionic Kähler manifold is of even dimension

n = 2 s \geq 2

and we can choose a canonical orthonormal local frame, called an adapted slant frame, as follows:

{e_{1}, e_{2} = sec θ P_{α} e_{1}, \dots, e_{2 s - 1}, e_{2 s} = sec θ P_{α} e_{2 s - 1}}

, where α is 1, 2 or 3 (see [41]).

3 Proof of Theorem 1.1

(i)

Since

{\bar{M}}^{4 m} (c)

is a quaternionic space form, from (6) and Gauss equation (5) we can easily obtain:

n^{2} {∥ H ∥}^{2} = 2 τ (p) + {∥ h ∥}^{2} - \frac{n (n - 1) c}{4} - \frac{3 c}{4} \sum_{β = 1}^{3} \sum_{i, j = 1}^{n} g^{2} (P_{β} e_{i}, e_{j}) .

(9)

Choosing now an adapted slant basis

{e_{1}, e_{2} = sec θ P_{α} e_{1}, \dots, e_{2 s - 1}, e_{2 s} = sec θ P_{α} e_{2 s - 1}}

T_{p} M

p \in M

, where

2 s = n

and making use of (7) and (8), we derive

g^{2} (P_{β} e_{i}, e_{i + 1}) = g^{2} (P_{β} e_{i + 1}, e_{i}) = {cos}^{2} θ for i = 1, 3, \dots, 2 s - 1

(10)

and

g (P_{β} e_{i}, e_{j}) = 0 for (i, j) \notin {(2 l - 1, 2 l), (2 l, 2 l - 1) | l \in {1, 2, \dots, s}} .

(11)

From (9), (10), and (11) we deduce that

2 τ (p) = n^{2} {∥ H ∥}^{2} - n C + \frac{c}{4} [n (n - 1) + 9 n {cos}^{2} θ] .

(12)

We define now the following function, denoted by

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2014-123/MediaObjects/13660_2013_Article_1329_IEq69_HTML.gif

, which is a quadratic polynomial in the components of the second fundamental form:

P = \frac{1}{2} n (n - 1) C + \frac{1}{2} (n + 1) C (L) - 2 τ (p) + \frac{c}{4} [n (n - 1) + 9 n {cos}^{2} θ],

where L is a hyperplane of

T_{p} M

. We can assume without loss of generality that L is spanned by

e_{1}, \dots, e_{n - 1}

. Then we have

\begin{array}{rcl} P & = & \frac{n - 1}{2} \sum_{α = n + 1}^{4 m} \sum_{i, j = 1}^{n} {(h_{i j}^{α})}^{2} + \frac{n + 1}{2 (n - 1)} \sum_{α = n + 1}^{4 m} \sum_{i, j = 1}^{n - 1} {(h_{i j}^{α})}^{2} \\ - 2 τ (p) + \frac{c}{4} [n (n - 1) + 9 n {cos}^{2} θ] . \end{array}

(13)

From (12) and (13), we derive

\begin{array}{rcl} P & = & \frac{n + 1}{2} \sum_{α = n + 1}^{4 m} \sum_{i, j = 1}^{n} {(h_{i j}^{α})}^{2} \\ + \frac{n + 1}{2 (n - 1)} \sum_{α = n + 1}^{4 m} \sum_{i, j = 1}^{n - 1} {(h_{i j}^{α})}^{2} - \sum_{α = n + 1}^{4 m} {(\sum_{i = 1}^{n} h_{i i}^{α})}^{2} \end{array}

and now we obtain easily that

\begin{array}{rcl} P & = & \sum_{α = n + 1}^{4 m} \sum_{i = 1}^{n - 1} [\frac{n^{2} - n + 2}{2 (n - 1)} {(h_{i i}^{α})}^{2} + (n + 1) {(h_{i n}^{α})}^{2}] \\ + \sum_{α = n + 1}^{4 m} [\frac{n (n + 1)}{n - 1} \sum_{i < j = 1}^{n - 1} {(h_{i j}^{α})}^{2} - 2 \sum_{i < j = 1}^{n} h_{i i}^{α} h_{j j}^{α} + \frac{n - 1}{2} {(h_{n n}^{α})}^{2}] . \end{array}

(14)

From (14) it follows that the critical points

h^{c} = (h_{11}^{n + 1}, h_{12}^{n + 1}, \dots, h_{n n}^{n + 1}, \dots, h_{11}^{4 m}, h_{12}^{4 m}, \dots, h_{n n}^{4 m})

are the solutions of the following system of linear homogeneous equations:

{\begin{matrix} \frac{\partial P}{\partial h_{i i}^{α}} = \frac{n (n + 1)}{n - 1} h_{i i}^{α} - 2 \sum_{k = 1}^{n} h_{k k}^{α} = 0, \\ \frac{\partial P}{\partial h_{n n}^{α}} = (n - 1) h_{n n}^{α} - 2 \sum_{k = 1}^{n - 1} h_{k k}^{α} = 0, \\ \frac{\partial P}{\partial h_{i j}^{α}} = \frac{2 n (n + 1)}{n - 1} h_{i j}^{α} = 0, \\ \frac{\partial P}{\partial h_{i n}^{α}} = 2 (n + 1) h_{i n}^{α} = 0, \end{matrix}

(15)

with

i, j \in {1, \dots, n - 1}

i \neq j

, and

α \in {n + 1, \dots, 4 m}

From (15) it follows that every solutions

h^{c}

has

h_{i j}^{α} = 0

for

i \neq j

and the determinant which corresponds to the first two sets of equations of the above system is zero (there exist solutions for non-totally geodesic submanifolds). Moreover, the Hessian matrix of

has the eigenvalues

\begin{matrix} λ_{11} = 0, λ_{22} = \frac{2 n^{2} - 5 n + 5}{n - 1}, λ_{33} = \dots = λ_{n n} = \frac{n (n + 1)}{n - 1}, \\ λ_{i j} = \frac{2 n (n + 1)}{n - 1}, λ_{i n} = 2 (n + 1), \forall i, j \in {1, \dots, n - 1}, i \neq j . \end{matrix}

Thus, it follows that

is parabolic and reaches a minimum

P (h^{c})

for each solution

h^{c}

of the system (15). But inserting (15) in (14) we obtain

P (h^{c}) = 0

. So

P \geq 0

, and this implies

2 τ (p) \leq \frac{1}{2} n (n - 1) C + \frac{1}{2} (n + 1) C (L) + \frac{c}{4} [n (n - 1) + 9 n {cos}^{2} θ] .

Hence we deduce that

ρ \leq \frac{1}{2} C + \frac{n + 1}{2 n (n - 1)} C (L) + \frac{c}{4} [1 + \frac{9}{n - 1} {cos}^{2} θ]

for every tangent hyperplane L of M. Taking now the infimum over all tangent hyperplane L we obtain (1).

Moreover, we can easily see now that the equality sign holds in the inequality (1) if and only if

h_{i j}^{α} = 0, \forall i, j \in {1, \dots, n}, i \neq j and α \in {n + 1, \dots, 4 m}

(16)

and

h_{n n}^{α} = 2 h_{11}^{α} = 2 h_{22}^{α} = \dots = 2 h_{n - 1, n - 1}^{α}, \forall α \in {n + 1, \dots, 4 m} .

(17)

From (16) and (17) we conclude that the equality sign holds in the inequality (1) if and only if the submanifold M is invariantly quasi-umbilical with trivial normal connection in

\bar{M}

, such that with respect to suitable orthonormal tangent and normal orthonormal frames, the shape operators take the forms (2).

(ii)

can be proved in a similar way, considering the following quadratic polynomial in the components of the second fundamental form:

Q = 2 n (n - 1) C - \frac{1}{2} (2 n - 1 (n - 1)) C (L) - 2 τ (p) + \frac{c}{4} [n (n - 1) + 9 n {cos}^{2} θ],

where L is a hyperplane of

T_{p} M

Acknowledgements

The third author was supported by CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0118.

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Vorheriger Artikel A fixed point theorem for preordered complete fuzzy quasi-metric spaces and an application

Nächster Artikel Commutators of intrinsic square functions on generalized Morrey spaces

Chen B-Y: Some pinching and classification theorems for minimal submanifolds. Arch. Math. 1993, 60: 568-578.CrossRef

Chen B-Y: δ -Invariants, inequalities of submanifolds and their applications. In Topics in Differential Geometry. Ed. Acad. Române, Bucharest; 2008:29-155.

Deng S: Improved Chen-Ricci inequality for Lagrangian submanifolds in quaternion space forms. Int. Electron. J. Geom. 2012,5(1):163-170.MathSciNet

Hong Y, Houh CS: Lagrangian submanifolds of quaternion Kaehlerian manifolds satisfying Chen’s equality. Beitr. Algebra Geom. 1998,39(2):413-421.MathSciNet

Liu X: On Ricci curvature of totally real submanifolds in a quaternion projective space. Arch. Math. 2002,38(4):297-305.MathSciNet

Liu X, Dai W: Ricci curvature of submanifolds in a quaternion projective space. Commun. Korean Math. Soc. 2002,17(4):625-633.MathSciNetCrossRef

Mihai I, Al-Solamy FR, Shahid MH: On Ricci curvature of a quaternion CR-submanifold in a quaternion space form. Rad. Mat. 2003,12(1):91-98.MathSciNet

Shahid MH, Al-Solamy FR: Ricci tensor of slant submanifolds in a quaternion projective space. C. R. Math. Acad. Sci. Paris 2011,349(9):571-573.MathSciNetCrossRef

Shukla SS, Rao PK: Ricci curvature of quaternion slant submanifolds in quaternion space forms. Acta Math. Acad. Paedagog. Nyházi 2012,28(1):69-81.MathSciNet

10.

Vîlcu GE: Slant submanifolds of quaternionic space forms. Publ. Math. (Debr.) 2012,81(3-4):397-413.CrossRef

11.

Yoon DW: A basic inequality of submanifolds in quaternionic space forms. Balk. J. Geom. Appl. 2004,9(2):92-102.

12.

Vîlcu GE: On Chen invariant and inequalities in quaternionic geometry. J. Inequal. Appl. 2013., 2013: Article ID 66

13.

Alegre P, Chen B-Y, Munteanu M: Riemannian submersions, δ -invariants, and optimal inequality. Ann. Glob. Anal. Geom. 2012,42(3):317-331.MathSciNetCrossRef

14.

Al-Solamy FR, Chen B-Y, Deshmukh S: Two optimal inequalities for anti-holomorphic submanifolds and their applications. Taiwan. J. Math. 2014,18(1):199-217.MathSciNetCrossRef

15.

Chen B-Y: An optimal inequality for CR-warped products in complex space forms involving CR δ -invariant. Int. J. Math. 2012.,23(3): Article ID 1250045

16.

Chen B-Y, Dillen F, Van der Veken J, Vrancken L: Curvature inequalities for Lagrangian submanifolds: the final solution. Differ. Geom. Appl. 2013,31(6):808-819.MathSciNetCrossRef

17.

Chen B-Y, Prieto-Martín A, Wang X:Lagrangian submanifolds in complex space forms satisfying an improved equality involving

δ (2, 2)

. Publ. Math. (Debr.) 2013,82(1):193-217.CrossRef

18.

Gülbahar M, Kiliç E, Keleş S: Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold. J. Inequal. Appl. 2013., 2013: Article ID 266

19.

Gülbahar M, Kiliç E, Keleş S: Some inequalities on screen homotethic lightlike hypersurfaces of a Lorentzian manifold. Taiwan. J. Math. 2013,17(6):2083-2100.CrossRef

20.

Özgür C, Mihai A: Chen inequalities for submanifolds of real space forms with a semi-symmetric non-metric connection. Can. Math. Bull. 2012,55(3):611-622.CrossRef

21.

Al-Solamy FR, Khan MA: Semi-invariant warped product submanifolds of almost contact manifolds. J. Inequal. Appl. 2012., 2012: Article ID 127

22.

Al-Solamy FR, Khan MA: Warped product submanifolds of Riemannian product manifolds. Abstr. Appl. Anal. 2012., 2012: Article ID 724898

23.

Al-Solamy FR, Khan MA: Semi-slant warped product submanifolds of a Kenmotsu manifold. Math. Probl. Eng. 2012., 2012: Article ID 708191

24.

Khan MA, Uddin S, Sachdeva R: Semi-invariant warped product submanifolds of cosymplectic manifolds. J. Inequal. Appl. 2012., 2012: Article ID 19

25.

Uddin S, Khan KA: An inequality for contact CR-warped product submanifolds of nearly cosymplectic manifolds. J. Inequal. Appl. 2012., 2012: Article ID 304

26.

Casorati F: Mesure de la courbure des surfaces suivant l’idée commune. Ses rapports avec les mesures de courbure gaussienne et moyenne. Acta Math. 1890,14(1):95-110.MathSciNetCrossRef

27.

Albertazzi L: Handbook of Experimental Phenomenology: Visual Perception of Shape, Space and Appearance. Wiley, Chichester; 2013.CrossRef

28.

Haesen S, Kowalczyk D, Verstraelen L: On the extrinsic principal directions of Riemannian submanifolds. Note Mat. 2009,29(2):41-53.MathSciNet

29.

Verstraelen L: The geometry of eye and brain. Soochow J. Math. 2004,30(3):367-376.MathSciNet

30.

Verstraelen L: Geometry of submanifolds I. The first Casorati curvature indicatrices. Kragujev. J. Math. 2013,37(1):5-23.MathSciNet

31.

Decu S, Haesen S, Verstraelen L: Optimal inequalities involving Casorati curvatures. Bull. Transylv. Univ. Braşov, Ser. B 2007,14(49):85-93. suppl.MathSciNet

32.

Decu S, Haesen S, Verstraelen L: Optimal inequalities characterising quasi-umbilical submanifolds. J. Inequal. Pure Appl. Math. 2008.,9(3): Article ID 79

33.

Ghişoiu V: Inequalities for the Casorati curvatures of slant submanifolds in complex space forms. In Riemannian Geometry and Applications. Proceedings RIGA 2011. Ed. Univ. Bucureşti, Bucharest; 2011:145-150.

34.

Chen B-Y: Pseudo-Riemannian Geometry, δ-Invariants and Applications. World Scientific, Hackensack; 2011.CrossRef

35.

Blair D: Quasi-umbilical, minimal submanifolds of Euclidean space. Simon Stevin 1977, 51: 3-22.MathSciNet

36.

Ishihara S: Quaternion Kählerian manifolds. J. Differ. Geom. 1974, 9: 483-500.MathSciNet

37.

Gray A:A note on manifolds whose holonomy group is a subgroup of

S p (n) \cdot S p (1)

. Mich. Math. J. 1969, 16: 125-128.CrossRef

38.

Chen B-Y, Houh CS: Totally real submanifolds of a quaternion projective space. Ann. Mat. Pura Appl. 1979,120(1):185-199.MathSciNetCrossRef

39.

Şahin B: Slant submanifolds of quaternion Kaehler manifolds. Commun. Korean Math. Soc. 2007,22(1):123-135.MathSciNetCrossRef

40.

Barros M, Chen B-Y, Urbano F: Quaternion CR-submanifolds of quaternion manifolds. Kodai Math. J. 1981, 4: 399-417.MathSciNetCrossRef

41.

Vîlcu GE: B.-Y. Chen inequalities for slant submanifolds in quaternionic space forms. Turk. J. Math. 2010,34(1):115-128.

Titel: Inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms
verfasst von: Vladimir Slesar
Bayram Şahin
Gabriel-Eduard Vîlcu
Publikationsdatum: 01.12.2014
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2014
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2014-123

Springer Professional

Inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Preliminaries

2.1 Riemannian invariants

2.2 Quaternionic Kähler manifolds

3 Proof of Theorem 1.1

Acknowledgements

Competing interests

Authors’ contributions

Premium Partner

Springer Professional

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Preliminaries

2.1 Riemannian invariants

2.2 Quaternionic Kähler manifolds

3 Proof of Theorem 1.1

Acknowledgements

Competing interests

Authors’ contributions

Weitere Artikel der Ausgabe 1/2014

Asymptotic normality of Huber-Dutter estimators in a linear EV model with AR(1) processes

Boundedness of Marcinkiewicz integrals with mixed homogeneity along compound surfaces

A new iteration method for variational inequalities on the set of common fixed points for a finite family of quasi-pseudocontractions in Hilbert spaces

Approximation properties of bivariate complex q-Balàzs-Szabados operators of tensor product kind

Convergence theorems for continuous pseudocontractive mappings in Banach spaces

A class of generalized pseudo-splines

Premium Partner