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

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 $h(X,JY)=h(JX,Y)$ 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

$${\pi }_1({\nabla }_{X}PY-{\nabla }_{Y}PX)={\pi }_1(A_{FY}X-A_{FX}Y),$$

(18)

for all $X,Y\in D_2$, where ${\pi }_1$ 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 $\left({\nabla }_{X}P\right)Y=0$, 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 $\left({\nabla }_{X}F\right)Y=0$, and $\left({\nabla }_{X}P\right)Y=A_{FY}X$ 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 $X\in D_i$, $N\in T^{\perp }M$, $i=1,2$.

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 $P{A}_{N}X=A_{N}PX$ for all $X\in D_T$ and $N\in T^{\perp }M$.

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 $\left({\nabla }_{X}P\right)Y=A_{FY}X$ and $\left({\nabla }_{X}F\right)Y=0$ 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 $A_{FX}Y=A_{FY}X$ for all $X,Y\in D_{\perp }$.

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:

$${\pi }_1({\nabla }_{X}PY-{\nabla }_{Y}PX)={\pi }_1(A_{FY}X-A_{FX}Y),$$

(24)

for all $X,Y\in D_2$, where ${\pi }_1$ is the projection over the totally real distribution $D_{\perp }$.

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 $D_{\perp }$ is totally geodesic if and only if $\left({\nabla }_{X}F\right)Y=0$, and $P{\nabla }_{X}Y=-A_{FY}X$ for any $X,Y\in D_{\perp }$.

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 $\left({\nabla }_{X}F\right)Y=0$, and $P{\nabla }_{X}Y=-A_{FY}X$ 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 $\left({\nabla }_{X}F\right)Y=0$, and $P{\nabla }_{X}Y=-A_{FY}X$ for both any $X,Y\in D_{\perp }$ 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 $D_{\perp }-D_2$-mixed totally geodesic if and only if $\left({\nabla }_{X}P\right)Y=A_{FY}X$ and $\left({\nabla }_{X}F\right)Y=0$, 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.