Correction: Alegre, P.; Carriazo, A. Bi-Slant Submanifolds of Para Hermitian Manifolds. Mathematics 2019, 7, 618

Pablo Alegre; Alfonso Carriazo

doi:10.3390/math13071170

and

Departamento de Geometría y Topología, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain

^*

Author to whom correspondence should be addressed.

Mathematics2025, 13(7), 1170;https://doi.org/10.3390/math13071170

Version Notes

Order Reprints

In the published publication [1], all the Propositions were named as proofs. There has been also an update regarding Pablo Alegre’s affiliations and email; the correct ones are Departamento de Geometría y Topología, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain, and palegre@us.es.

The authors wish to make corrections to the labels of these Proposition. All the proofs were correct, so they are ommited:

5. Semi-Slant Submanifolds of a Para Kaehler Manifold

It is always interesting to study the integrability of the involved distributions.

Proposition 1.

Let M be a semi-slant submanifold of a para Hermitian manifold. Both the holomorphic and the slant distributions are P invariant.

Theorem 6.

Let M be a semi-slant submanifold of a para Kaehler manifold. The holomorphic distribution is integrable if and only if

h (X, J Y) = h (J X, Y)

for all

X, Y \in D_{T}

.

Theorem 7.

Let M be a semi-slant submanifold of a para Kaehler manifold. The slant distribution is integrable if and only if

π_{1} (\nabla_{X} P Y - \nabla_{Y} P X) = π_{1} (A_{F Y} X - A_{F X} Y),

(18)

for all

X, Y \in D_{2}

, where

π_{1}

is the projection over the invariant distribution

D_{T}

.

Now, we study the conditions for the involved distributions being totally geodesic.

Proposition 2.

Let M be a semi-slant submanifold of a para Kaehler manifold

\tilde{M}

. If the holomorphic distribution

D_{T}

is totally geodesic, then

(\nabla_{X} P) Y = 0

, and

\nabla_{X} Y \in D_{T}

for any

X, Y \in D_{T}

.

Proposition 3.

Let M be a semi-slant submanifold of a para Kaehler manifold

\tilde{M}

. The slant distribution

D_{2}

is totally geodesic if and only if

(\nabla_{X} F) Y = 0

, and

(\nabla_{X} P) Y = A_{F Y} X

for any

X, Y \in D_{2}

.

Given two orthogonal distributions

D_{1}

and

D_{2}

over a submanifold, it is called a

D_{1} - D_{2}

-mixed totally geodesic if

h (X, Y) = 0

for all

X \in D_{1}

,

Y \in D_{2}

.

Proposition 4.

Let M be a semi-slant submanifold of a para Hermitian manifold

\tilde{M}

. M is a mixed totally geodesic if and only if

A_{N} X \in D_{i}

for any

X \in D_{i}

,

N \in T^{⊥} M

,

i = 1, 2

.

Proposition 5.

Let M be a semi-slant submanifold of a para Kaehler manifold

\tilde{M}

. If

\nabla F = 0

, then either M is

D_{T} - D_{2}

-mixed totally geodesic or

h (X, Y)

is a eigenvector of

f^{2}

associated with the eigenvalue of one, for all

X \in D_{T}

,

Y \in D_{2}

.

Proposition 6.

Let M be a mixed totally geodesic semi-slant submanifold of a para Kaehler manifold

\tilde{M}

. If

D_{T}

is integrable, then

P A_{N} X = A_{N} P X

for all

X \in D_{T}

and

N \in T^{⊥} M

.

Finally, the mixed totally geodesic characterization can be summarized with:

Theorem 8.

Let M be a proper semi-slant submanifold of a para Kaehler manifold

\tilde{M}

. M is a

D_{T} - D_{2}

-mixed totally geodesic if and only if

(\nabla_{X} P) Y = A_{F Y} X

and

(\nabla_{X} F) Y = 0

for all

X, Y

in different distributions.

6. Hemi-Slant Submanifolds of a Para Kaehler Manifold

We will also study the integrability of the involved distributions for a hemi-slant submanifold.

Proposition 7.

Let M be a hemi-slant submanifold of a para Hermitian manifold. The slant distribution is P invariant.

Lemma 1.

Let M be a hemi-slant submanifold of a para Kaehler manifold. The totally real distribution is integrable if and only if

A_{F X} Y = A_{F Y} X

for all

X, Y \in D_{⊥}

.

The following result is known for hemi-slant submanifolds of Kaehler manifolds [14]. We obtain the equivalent one for hemi-slant submanifolds of para Kaehler manifolds:

Theorem 9.

Let M be a hemi-slant submanifold of a para Kaehler manifold. The totally real distribution is always integrable.

Now, we study the integrability of the slant distribution.

Theorem 10.

Let M be a hemi-slant submanifold of a para Kaehler manifold. The slant distribution is integrable if and only if:

π_{1} (\nabla_{X} P Y - \nabla_{Y} P X) = π_{1} (A_{F Y} X - A_{F X} Y),

(24)

for all

X, Y \in D_{2}

, where

π_{1}

is the projection over the totally real distribution

D_{⊥}

.

The proof is analogous to the one of Theorem 7.

Lemma 2.

Let M be a hemi-slant submanifold of a para Kaehler manifold

\tilde{M}

. The totally real distribution

D_{⊥}

is totally geodesic if and only if

(\nabla_{X} F) Y = 0

, and

P \nabla_{X} Y = - A_{F Y} X

for any

X, Y \in D_{⊥}

.

The same proof of Proposition 3 is valid for the slant distribution of a hemi-slant distribution.

Lemma 3.

Let M be a hemi-slant submanifold of a para Kaehler manifold

\tilde{M}

. The slant distribution

D_{2}

is totally geodesic if and only if

(\nabla_{X} F) Y = 0

, and

P \nabla_{X} Y = - A_{F Y} X

for any

X, Y \in D_{2}

.

Remember that the classical De Rham–Wu Theorem [18,19], says that two orthogonal, complementary, and geodesic foliations (called a direct product structure) in a complete and simply connected semi-Riemannian manifold give rise to a global decomposition as a direct product of two leaves. Therefore, from the previous lemmas, it is directly deduced:

Remark 5.

Let M be a complete and simply-connected hemi-slant submanifold of a para Kaehler manifold

\tilde{M}

. Then, M is locally the product of the integral submanifolds of the slant distributions if and only if

(\nabla_{X} F) Y = 0

, and

P \nabla_{X} Y = - A_{F Y} X

for both any

X, Y \in D_{⊥}

and

X, Y \in D_{2}

.

Finally, we can also study when a hemi-slant submanifold is mixed totally geodesic. We get a result similar to Theorem 8, but now the proof is much easier.

Proposition 8.

Let M be a hemi-slant submanifold of a para Kaehler manifold

\tilde{M}

. M is a

D_{⊥} - D_{2}

-mixed totally geodesic if and only if

(\nabla_{X} P) Y = A_{F Y} X

and

(\nabla_{X} F) Y = 0

, for all

X, Y

in different distributions.

The authors state that the scientific conclusions are unaffected. This correction was ap-proved by the Academic Editor. The original publication has also been updated.

Reference

Alegre, P.; Carriazo, A. Bi-Slant Submanifolds of Para Hermitian Manifolds. Mathematics 2019, 7, 618. [Google Scholar] [CrossRef]

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Correction: Alegre, P.; Carriazo, A. Bi-Slant Submanifolds of Para Hermitian Manifolds. Mathematics 2019, 7, 618

5. Semi-Slant Submanifolds of a Para Kaehler Manifold

6. Hemi-Slant Submanifolds of a Para Kaehler Manifold

Reference

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