A study on curvature relations of conformal generic submersions

Definition 1

(Baird and Wood, 2003) “ Let $(\mathbb{M},g)$ and $(\mathbb{N},h)$ be two RMs with dimensions m and n, respectively, and let $\mathrm{\Pi }:(\mathbb{M},g)\to (\mathbb{N},h)$ be a $C^{\infty }$ – differentiable map between them. If either

1. ${\mathrm{\Pi }}_{*p}=0$ , or

2. ${\mathrm{\Pi }}_{*p}$ maps horizontal space ${\mathcal{H}}_p={\left(\mathit{ker}{\mathrm{\Pi }}_{*p}\right)}^{\perp }$ conformally onto $T_{\mathrm{\Pi }\left(p\right)}\mathbb{N}$ , i.e., ${\mathrm{\Pi }}_{*p}$ is surjective and there exists a number $\mathrm{\Lambda }\left(p\right)\ne 0$ such that

   (1)
   $$h({\mathrm{\Pi }}_{*p}\mathbb{X},{\mathrm{\Pi }}_{*p}\mathbb{Y})=\mathrm{\Lambda }\left(p\right)g(\mathbb{X},\mathbb{Y}),$$ for any $\mathbb{X},\mathbb{Y}\in \mathrm{\Gamma }{\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)}^{\perp }$ .

Then, $\mathrm{\Pi }$ is called horizontally weakly conformal or semi conformal at $p\in \mathbb{M}$ . ”.

Proposition 1

(Gudmundsson, 1992) “Let $\mathrm{\Pi }:(\mathbb{M},g)\to (\mathbb{N},h)$ be a horizontally conformal submersion with dilation $\lambda$ $\mathbb{X},\mathbb{Y}$ be horizontal vectors fields, then

$${\mathcal{A}}_{\mathbb{X}}\mathbb{Y}=\frac{1}{2}\left\{\mathcal{V}\right[\mathbb{X},\mathbb{Y}]-{\lambda }^{2}g(\mathbb{X},\mathbb{Y}\left){\mathbb{G}}_{\mathcal{V}}\right(\frac{1}{{\lambda }^2}\left)\right\}.\mathrm{”}$$

Theorem 1

“Let $({\mathbb{M}}^m,g,\nabla ,R)$ and $({\mathbb{N}}^n,h,\nabla ,R^{\ast })$ be two RMs, where $m>n⩾2$ . Let R and $R^{\ast }$ be the curvature tensors on $\mathbb{M}$ and $\mathbb{N}$ , respectively. Let $\mathrm{\Pi }:(\mathbb{M},g)\to (\mathbb{N},h)$ be a horizontally conformal submersion, with dilation $\lambda :\mathbb{M}\to {\mathbb{R}}^{+}$ and let $\widehat{R}$ be the curvature tensor of the fibres of the submersion. If $\mathbb{X},\mathbb{Y},\mathbb{Z},\mathbb{H}$ are horizontal $\mathbb{U},\mathbb{V},\mathbb{W},\mathbb{G}$ vertical vectors, then

$$g\left(R\right(\mathbb{U},\mathbb{V})\mathbb{W},\mathbb{G})=g\left(\widehat{R}\right(\mathbb{U},\mathbb{V})\mathbb{W},\mathbb{G})+g({\mathcal{T}}_{\mathbb{U}}\mathbb{W},{\mathcal{T}}_{\mathbb{V}}\mathbb{G})-g({\mathcal{T}}_{\mathbb{V}}\mathbb{W},{\mathcal{T}}_{\mathbb{U}}\mathbb{G}),$$

(10)

$$g\left(R\right(\mathbb{U},\mathbb{V})\mathbb{W},\mathbb{X})=\mathbb{g}({\left({\nabla }_{\mathbb{U}}\mathbb{T}\right)}_{\mathbb{V}}\mathbb{W},\mathbb{X})-\mathbb{g}({\left({\nabla }_{\mathbb{V}}\mathbb{T}\right)}_{\mathbb{U}}\mathbb{W},\mathbb{X}),$$

(11)

$$g\left(\mathbb{R}\right(\mathbb{U},\mathbb{X})\mathbb{Y},\mathbb{V})=g({\left({\nabla }_{\mathbb{U}}A\right)}_{\mathbb{X}}\mathbb{Y},\mathbb{V})+g(A_{\mathbb{X}}\mathbb{U},A_{\mathbb{Y}}\mathbb{V})$$ $$\begin{array}{c}\hfill -g({\left({\nabla }_{\mathbb{X}}T\right)}_{\mathbb{U}}\mathbb{Y},\mathbb{V})-g({\mathcal{T}}_{\mathbb{V}}\mathbb{Y},{\mathcal{T}}_{\mathbb{U}}\mathbb{X})\\ \hfill +{\lambda }^{2}g(A_{\mathbb{X}}\mathbb{Y},\mathbb{U})g(\mathbb{V},{\mathbb{G}}_{\mathcal{V}}(\frac{1}{{\lambda }^2}\left)\right),\end{array}$$

(12)

$$g\left(R\right(\mathbb{X},\mathbb{Y})\mathbb{Z},\mathbb{H})=\frac{1}{{\lambda }^2}h\left(R^{\ast }\right({\mathbb{X}}_{\ast },{\mathbb{Y}}_{\ast }),{\mathbb{Z}}_{\ast },{\mathbb{H}}_{\ast })+\frac{1}{4}\left\{g\left(\mathcal{V}\right[\mathbb{X},\mathbb{Z}],\mathcal{V}[\mathbb{Y},H\left]\right)\right.$$ $$\begin{array}{c}\hfill \left.-g\left(\mathcal{V}\right[\mathbb{Y},\mathbb{Z}],\mathcal{V}[\mathbb{X},\mathbb{H}\left]\right)+2g\left(\mathcal{V}\right[\mathbb{X},\mathbb{Y}],\mathcal{V}[\mathbb{Z},\mathbb{H}\left]\right)\right\}\\ \hfill +\frac{{\lambda }^2}{2}\left\{g(\mathbb{X},\mathbb{Z})g\left({\nabla }_{\mathbb{Y}}\mathbb{G}\right(\frac{1}{{\lambda }^2}),H)-g(\mathbb{Y},\mathbb{Z})g\left({\nabla }_{\mathbb{X}}\mathbb{G}\right(\frac{1}{{\lambda }^2}),\mathbb{H})\right.\\ \hfill \left.+g(\mathbb{Y},\mathbb{H})g\left({\nabla }_{\mathbb{X}}\mathbb{G}\right(\frac{1}{{\lambda }^2}),\mathbb{Z})-g(\mathbb{X},\mathbb{H})g\left({\nabla }_{\mathbb{Y}}\mathbb{G}\right(\frac{1}{{\lambda }^2}),\mathbb{Z})\right\}\\ \hfill +\frac{{\lambda }^4}{4}\left\{\left(g\right(\mathbb{X},\mathbb{H}\left)g\right(\mathbb{Y},\mathbb{Z})-g(\mathbb{Y},\mathbb{H}\left)g\right(\mathbb{X},\mathbb{Z}\left)\right)\Vert \mathbb{G}\left(\frac{1}{{\lambda }^2}\right){\Vert }^2\right.\\ \hfill \left.+g\left(\mathbb{X}\right(\frac{1}{{\lambda }^2})\mathbb{Y}-\mathbb{Y}(\frac{1}{{\lambda }^2})\mathbb{X},\mathbb{H}(\frac{1}{{\lambda }^2})\mathbb{Z}-\mathbb{Z}(\frac{1}{{\lambda }^2}\left)H\right)\right\}.\mathrm{”}\end{array}$$

Lemma 1

(Urakawa, 1993) “Consider that $\phi :\mathbb{M}\to \mathbb{N}$ is a smooth map between the RMs $(\mathbb{M},g)$ and $(\mathbb{N},h)$ . Then

$${\nabla }_{\mathbb{X}}^{\phi }{\phi }_{\ast }\left(\mathbb{Y}\right)-{\nabla }_{\mathbb{Y}}^{\phi }{\phi }_{\ast }\left(\mathbb{X}\right)-{\phi }_{\ast }\left(\right[\mathbb{X},\mathbb{Y}\left]\right)=0,$$ for any $\mathbb{X},\mathbb{Y}\in \mathrm{\Gamma }\left(T\mathbb{M}\right)$ .”

Lemma 2

“Assume that the submersion $\mathrm{\Pi }:\mathbb{M}\to \mathbb{N}$ is horizontally conformal. Then, for any vertical fields $\mathbb{V},\mathbb{W}$ and horizontal vector fields $\mathbb{X},\mathbb{Y}$ ,

1. $\left(\nabla {\mathrm{\Pi }}_{\ast }\right)(\mathbb{X},\mathbb{Y})=\mathbb{X}\left(\ln \lambda \right){\mathrm{\Pi }}_{\ast }\mathbb{Y}+\mathbb{Y}\left(\ln \lambda \right){\mathrm{\Pi }}_{\ast }\mathbb{X}-g(\mathbb{X},\mathbb{Y}){\mathrm{\Pi }}_{\ast }\left(\mathbb{G}\ln \lambda \right)$ ;

2. $\left(\nabla {\mathrm{\Pi }}_{\ast }\right)(\mathbb{V},\mathbb{W})=-{\mathrm{\Pi }}_{\ast }\left({\mathcal{T}}_{\mathbb{V}}\mathbb{W}\right)$ ;

3. $\left(\nabla {\mathrm{\Pi }}_{\ast }\right)(\mathbb{X},\mathbb{V})=-{\mathrm{\Pi }}_{\ast }\left({\nabla }_{\mathbb{X}}\mathbb{V}\right)=-{\mathrm{\Pi }}_{\ast }\left(A_{\mathbb{X}}\mathbb{V}\right)$ .”

Definition 2

Let $(\mathbb{M},g,\mathcal{F})$ be a l.p.R. manifold with the product structure $\mathcal{F}$ and $(\mathbb{N},h)$ be a RM. Consider a horizontally conformal submersion $\mathrm{\Pi }:(\mathbb{M},g,\mathcal{F})\to (\mathbb{N},h)$ . Then $\mathrm{\Pi }$ is called CGS if there are two orthogonal complementary distributions $\mathbb{D}$ and ${\mathbb{D}}_1$ of $\ker {\mathrm{\Pi }}_{\ast }$ such that

$$\mathit{ker}{\mathrm{\Pi }}_{\ast }=\mathbb{D}\oplus {\mathbb{D}}_1,F\left(\mathbb{D}\right)=\mathbb{D},$$ where ${\left({\mathbb{D}}_1\right)}_p=\mathit{ker}{\mathrm{\Pi }}_{\ast }\cap F\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right),p\in \mathbb{M}$ , a complex subspace of a vertical space ${\mathcal{V}}_p$ , has a constant dimension along $\mathbb{M}$ and defines a differentiable distribution on $\mathbb{M}$ . The distribution ${\mathbb{D}}_1$ is called the purely real distribution.

Example 1

Let $\mathrm{\Pi }:{\mathbb{R}}^8\to {\mathbb{R}}^2$ be a submersion defined by $$\mathrm{\Pi }(u_1,u_2,u_3,u_4,u_5,u_6,u_7,u_8)=e^7\left(\frac{1}{\sqrt{2}}(u_3+u_7),\frac{1}{\sqrt{2}}(u_3-u_7)\right)\text{.}$$ Then it follows that $$\mathit{ker}{\mathrm{\Pi }}_{\ast }=\langle V_1=\frac{\partial }{\partial x_1},V_2=\frac{\partial }{\partial x_2},V_3=\frac{\partial }{\partial x_4},V_4=\frac{\partial }{\partial x_5},V_5=\frac{\partial }{\partial x_6},V_6=\frac{\partial }{\partial x_8}\rangle$$ and $${\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)}^{\perp }=\langle {\mathbb{X}}_1=\frac{1}{\sqrt{2}}\left(\frac{\partial }{\partial x_3}+\frac{\partial }{\partial x_7}\right),{\mathbb{X}}_2=\frac{1}{\sqrt{2}}\left(\frac{\partial }{\partial x_3}-\frac{\partial }{\partial x_7}\right)\rangle ,$$

Moreover, $$\begin{array}{cc}\hfill \mathcal{F}{V}_1=\frac{1}{\sqrt{2}}{V}_2-\frac{1}{\sqrt{2}}{V}_6,\mathcal{F}{V}_2& =-\frac{1}{\sqrt{2}}{V}_1-\frac{1}{2}{\mathbb{X}}_1+\frac{1}{2}{\mathbb{X}}_2,\hfill \\ \hfill \mathcal{F}{V}_3=\frac{1}{\sqrt{2}}{V}_4-\frac{1}{2}{\mathbb{X}}_1-\frac{1}{2}{\mathbb{X}}_2,{\mathit{FV}}_4& =-\frac{1}{\sqrt{2}}{V}_3+\frac{1}{\sqrt{2}}{V}_5,\hfill \\ \hfill \mathcal{F}{V}_5=\frac{1}{\sqrt{2}}{V}_4+\frac{1}{2}{\mathbb{X}}_1+\frac{1}{2}{\mathbb{X}}_2,\mathcal{F}{V}_6& =-\frac{1}{\sqrt{2}}{V}_1+\frac{1}{2}{\mathbb{X}}_1-\frac{1}{2}{\mathbb{X}}_2\text{.}\hfill \end{array}$$

Hence, $\mathbb{D}=V_1,V_4$ and ${\mathbb{D}}_1=V_2,V_3,V_5,V_6$ . Also by direct computations, we obtain $$g_{{\mathbb{R}}^8}({\mathbb{X}}_1,{\mathbb{X}}_1)={\left(e^7\right)}^{-2}{g}_{{\mathbb{R}}^2}({\mathrm{\Pi }}_{\ast }{\mathbb{X}}_1,{\mathrm{\Pi }}_{\ast }{\mathbb{X}}_1)\mathrm{and}{g}_{{\mathbb{R}}^8}({\mathbb{X}}_2,{\mathbb{X}}_2)={\left(e^7\right)}^{-2}{g}_{{\mathbb{R}}^2}({\mathrm{\Pi }}_{\ast }{\mathbb{X}}_2,{\mathrm{\Pi }}_{\ast }{\mathbb{X}}_2),$$ where $g_{{\mathbb{R}}^8}$ and $g_{{\mathbb{R}}^2}$ denote the standard metrices on ${\mathbb{R}}^8$ and ${\mathbb{R}}^2$ , respectively. Thus $\mathrm{\Pi }$ is CGS with $\lambda =e^7$ .

Proposition 2

Let $\mathrm{\Pi }:(\mathbb{M},g,\mathcal{F})\to (\mathbb{N},h)$ be a CGS from a l.p.R. manifold $(\mathbb{M},g,\mathcal{F})$ onto a RM $(\mathbb{N},h)$ . Then $$\begin{array}{c}\hfill \left(i\right)\alpha \mathbb{D}=\mathbb{D},\left(\mathit{ii}\right)\beta {\mathbb{D}}_1=0,\left(\mathit{iii}\right)\alpha {\mathbb{D}}_1\subset {\mathbb{D}}_1,\left(\mathit{iv}\right)B{\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)}^{\perp }={\mathbb{D}}_1,\\ \hfill \left(v\right){\alpha }^2+B\beta =\mathit{id},\left(\mathit{vi}\right)C^2+\beta B=\mathit{id},\left(\mathit{vii}\right)\beta \alpha +C\beta =0,\left(\mathit{viii}\right)\mathit{BC}+\alpha B=0\text{.}\end{array}$$

Proof

These can easily be obtained with the help of (19)–(21).

Lemma 3

Let $\mathrm{\Pi }:(\mathbb{M},g,\mathcal{F})\to (B,h)$ be a CGS from a l.p.R. manifold $(\mathbb{M},g,\mathcal{F})$ onto a RM $(B,h)$ . Then

1. ${\mathcal{A}}_{\mathbb{X}}B\mathbb{Y}+\mathcal{H}{\nabla }_{\mathbb{X}}C\mathbb{Y}=C\mathcal{H}{\nabla }_{\mathbb{X}}\mathbb{Y}+\beta {\mathcal{A}}_{\mathbb{X}}\mathbb{Y}$

   $\mathcal{V}{\nabla }_{\mathbb{X}}B\mathbb{Y}+{\mathcal{A}}_{\mathbb{X}}C\mathbb{Y}=B\mathcal{H}{\nabla }_{\mathbb{X}}\mathbb{Y}+\alpha {\mathcal{A}}_{\mathbb{X}}\mathbb{Y}$ ,

2. ${\mathcal{T}}_U\alpha V+{\mathcal{A}}_{\beta V}U=C{\mathcal{T}}_{U}V+\beta {\widehat{\nabla }}_{U}V$

   ${\widehat{\nabla }}_U\alpha V+{\mathcal{T}}_U\beta V=B{\mathcal{T}}_{U}V+\alpha {\widehat{\nabla }}_{U}V$ ,

3. ${\mathcal{A}}_{\mathbb{X}}\alpha U+\mathcal{H}{\nabla }_U\beta V=C{\mathcal{A}}_{\mathbb{X}}U+\beta \mathcal{V}{\nabla }_{\mathbb{X}}U$

   $\mathcal{V}{\nabla }_{\mathbb{X}}\alpha U+{\mathcal{A}}_{\mathbb{X}}\beta U=B{\mathcal{A}}_{\mathbb{X}}U+\alpha \mathcal{V}{\nabla }_{\mathbb{X}}U$ ,

for any $U,V\in \mathrm{\Gamma }\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)$ and $\mathbb{X},\mathbb{Y}\in \mathrm{\Gamma }{\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)}^{\perp }$ .

Theorem 2

Let $\mathrm{\Pi }:(\mathbb{M},g,\mathcal{F})\to (B,h)$ be a CGS from l.p.R. manifold $(M,g,\mathcal{F})$ onto a RM $(B,h)$ . Then the f.a.e;

1. The distribution $\mathbb{D}$ is integrable.

2. $h\left(\right(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_1,\mathcal{F}{U}_2)-(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_2,\mathcal{F}{U}_1),{\mathrm{\Pi }}_{\ast }\beta V)={\lambda }^{2}g\left(\alpha \right({\widehat{\nabla }}_{U_1}\mathcal{F}{U}_2-{\widehat{\nabla }}_{U_2}\mathcal{F}{U}_1),V),$

3. $({\mathcal{T}}_{U_2}\mathcal{F}{U}_1-{\mathcal{T}}_{U_1}\mathcal{F}{U}_2)\in \mathrm{\Gamma }\left(\mu \right)$ and $({\widehat{\nabla }}_{U_1}\mathcal{F}{U}_2-{\widehat{\nabla }}_{U_2}\mathcal{F}{U}_1)\in \mathrm{\Gamma }\left(\mathbb{D}\right)$ ,

for any $U_1,U_2\in \mathrm{\Gamma }\left(\mathbb{D}\right)$ and $V\in \mathrm{\Gamma }\left({\mathbb{D}}_1\right)$ .

Proof

For any $U_1,U_2\in \mathrm{\Gamma }\left(\mathbb{D}\right),V\in \mathrm{\Gamma }\left({\mathbb{D}}_1\right)$ , applying (16), (17), (4) and (19), we get $$g\left(\right[U_1,U_2],V)=g(\mathcal{H}{\nabla }_{U_1}\mathcal{F}{U}_2,\beta V)+g({\widehat{\nabla }}_{U_1}\mathcal{F}{U}_2,\alpha V)-g(\mathcal{H}{\nabla }_{U_2}\mathcal{F}{U}_1,\beta V)-g({\widehat{\nabla }}_{U_2}\mathcal{F}{U}_1,\alpha V)\text{.}$$

Since $\mathrm{\Pi }$ is a CGS, using (19) and Lemma 2, we get

$$\begin{array}{c}\hfill g\left(\right[U_1,U_2],V)={\lambda }^{-2}h(-(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_1,\mathcal{F}{U}_2)+{\nabla }_U^{\mathrm{\Pi }}{\mathrm{\Pi }}_{\ast }\mathcal{F}{U}_2,{\mathrm{\Pi }}_{\ast }\beta V)-{\lambda }^{-2}h(-(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_2,\mathcal{F}{U}_1)\\ \hfill +{\nabla }_V^{\mathrm{\Pi }}{\mathrm{\Pi }}_{\ast }\mathcal{F}{U}_1,{\mathrm{\Pi }}_{\ast }\beta V)+g(\mathcal{F}({\widehat{\nabla }}_{U_1}\mathcal{F}{U}_2-{\widehat{\nabla }}_{U_2}\mathcal{F}{U}_1),V)\\ \hfill ={\lambda }^{-2}h\left(\right(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_2,\mathcal{F}{U}_1)-(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_1,\mathcal{F}{U}_2),{\mathrm{\Pi }}_{\ast }\beta V)+g\left(\alpha \right({\widehat{\nabla }}_{U_1}\mathcal{F}{U}_2-{\widehat{\nabla }}_{U_2}\mathcal{F}{U}_1),V)\text{.}\end{array}$$

We note that $g\left(\right[U_1,U_2],W)=0$ , for any $W\in \mathrm{\Gamma }\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)$ as the distribution $\mathit{ker}{\mathrm{\Pi }}_{\ast }$ is always integrable. Therefore, $\left(a\right)\iff \left(b\right)$ follows. Moreover, with Lemma 2 and (1), (26) reduces to $$g\left(\right[U_1,U_2],V)=-g({\mathcal{T}}_{U_2}\mathcal{F}{U}_1-{\mathcal{T}}_{U_1}\mathcal{F}{U}_2,\beta V)+g\left(\alpha \right({\widehat{\nabla }}_{U_1}\alpha U_2-{\widehat{\nabla }}_{U_2}\alpha U_1),V)$$

By using (23) and Proposition 2, we obtain $\left(a\right)\iff \left(c\right)$ .

Theorem 3

Let $(\mathbb{M},g,\mathcal{F})$ be a l.p.R. manifold and $(B,h)$ , a RM and $\mathrm{\Pi }:(\mathbb{M},g,\mathcal{F})\to (B,h)$ be a CGS. Then the f.a.e;

1. The distribution ${\mathbb{D}}_1$ is integrable.

2. $h\left(\right(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(V_1,\alpha V_2)-(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(V_2,\alpha V_1),U)={\lambda }^{2}g({\mathcal{T}}_{V_1}\beta V_2-{\mathcal{T}}_{V_2}\beta V_1,U)$ ,

3. ${\widehat{\nabla }}_{V_1}\alpha V_2-{\widehat{\nabla }}_{V_2}\alpha V_1+{\mathcal{T}}_{V_1}\beta V_2-{\mathcal{T}}_{V_2}\beta V_1\in \mathrm{\Gamma }\left({\mathbb{D}}_2\right)$ ,

for any $U\in \mathrm{\Gamma }\left(\mathbb{D}\right)$ and $V_1,V_2\in \mathrm{\Gamma }\left({\mathbb{D}}_1\right)$ .

Proof

The distribution ${\mathbb{D}}_1$ to be integrable iff $g\left(\right[V_1,V_2],\mathcal{F}U)=0$ , as well as $g\left(\right[V_1,V_2],W)=0$ , for any $V_1,V_2\in \mathrm{\Gamma }\left({\mathbb{D}}_1\right),U\in \mathrm{\Gamma }\left(\mathbb{D}\right)$ and $W\in \mathrm{\Gamma }{\left(\ker {\mathrm{\Pi }}_{\ast }\right)}^{\perp }$ . As $\ker {\mathrm{\Pi }}_{\ast }$ is integrable, then we can easily get $g\left(\right[V_1,V_2],W)=0$ .

Moreover, by using (16), (17), (19) and (13), we obtain $$\begin{array}{cc}\hfill g\left(\right[V_1,V_2],\mathit{FU})& =g({\nabla }_{V_1}{\mathit{FV}}_2-{\nabla }_{V_2}{\mathit{FV}}_1,\mathit{FU})\hfill \\ \hfill & =g({\nabla }_{V_1}\alpha V_2,U)+g({\nabla }_{V_1}\beta V_2,U)-g({\nabla }_{V_2}\alpha V_1,U)-g({\nabla }_{V_2}\beta V_1,U)\text{.}\hfill \\ \hfill & =g({\widehat{\nabla }}_{V_1}\alpha V_2-{\widehat{\nabla }}_{V_2}\alpha V_1,U)+g({\mathcal{T}}_{V_1}\beta V_2-{\mathcal{T}}_{V_2}\beta V_1,U),\hfill \end{array}$$ which shows $\left(a\right)\iff \left(c\right)$ .

On the other side, applying (13), we arrive at $$g\left(\right[V_1,V_2],\mathcal{F}U)={\lambda }^{-2}h(-(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(V_1,\alpha V_2)+(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(V_2,\alpha V_1),U)+g({\mathcal{T}}_{V_1}\beta V_2-{\mathcal{T}}_{V_2}\beta V_1,U)\text{.}$$ Hence, $\left(a\right)\iff \left(b\right)$ .

Theorem 4

Let $\mathrm{\Pi }$ be a CGS from a l.p.R manifold $(\mathbb{M},g,\mathcal{F})$ to a RM $(\mathbb{N},h)$ i.e., $\mathrm{\Pi }:(\mathbb{M},g,\mathcal{F})\to (\mathbb{N},h)$ . Then, the distribution ${\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)}^{\perp }$ is integrable iff

1. There is no component of $\mathcal{V}({\nabla }_{\mathbb{X}}B\mathbb{Y}-{\nabla }_{\mathbb{Y}}B\mathbb{X})+{\mathcal{A}}_{\mathbb{X}}C\mathbb{Y}-{\mathcal{A}}_{\mathbb{Y}}C\mathbb{X}$ in $\mathrm{\Gamma }\left(\mathbb{D}\right)$ .

2. $$\begin{array}{c}\hfill {\lambda }^{2}g\left(\mathbb{Y}\left(\ln \lambda \right)C-\mathbb{X}\left(\ln \lambda \right)C\mathbb{Y}-\mathcal{C}\mathbb{Y}\left(\ln \lambda \right)\mathbb{X}+C\mathbb{X}\left(\ln \lambda \right)\mathbb{Y}+2g(\mathbb{X},C\mathbb{Y})\nabla \ln \lambda ,\beta U_2\right)\\ \hfill +g(-\alpha (\mathcal{V}{\nabla }_{\mathbb{X}}B\mathbb{Y}-\mathcal{V}{\nabla }_{\mathbb{Y}}B\mathbb{X}+{\mathcal{A}}_{\mathbb{X}}C\mathbb{Y}-{\mathcal{A}}_{\mathbb{Y}}C\mathbb{X}),U_2)\\ \hfill =h\left(\left(\nabla {\mathrm{\Pi }}_{\ast }\right)(\mathbb{X},B\mathbb{Y})-\left(\nabla {\mathrm{\Pi }}_{\ast }\right)(\mathbb{Y},B\mathbb{X})-{\nabla }_{\mathbb{X}}^{\mathrm{\Pi }}{\mathrm{\Pi }}_{\ast }\left(C\mathbb{Y}\right)+{\nabla }_{\mathbb{Y}}^{\mathrm{\Pi }}{\mathrm{\Pi }}_{\ast }\left(C\mathbb{X}\right),{\mathrm{\Pi }}_{\ast }\left(\beta U_2\right)\right),\end{array}$$

for $\mathbb{X},\mathbb{Y}\in \mathrm{\Gamma }\mathit{ker}{\mathrm{\Pi }}_{\ast }{\left.\right)}^{\perp },U_1\in \mathrm{\Gamma }\left(\mathbb{D}\right)$ and $U_2\in \mathrm{\Gamma }\left({\mathbb{D}}_1\right)$ .

Proof

Since the horizontal distribution is integrable if and only if $g\left(\right[\mathbb{X},\mathbb{Y}],W)=0$ , for any $\mathbb{X},\mathbb{Y}\in \mathrm{\Gamma }{\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)}^{\perp }$ and $W\in \mathrm{\Gamma }\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)$ , where $W=U_1+U_2$ such that $U_1\in \mathbb{D}$ and $U_2\in {\mathbb{D}}_1$ . This implies that $g\left(\right[\mathbb{X},\mathbb{Y}],U_1)=0$ and $g\left(\right[\mathbb{X},\mathbb{Y}],U_2)=0$ .

We omit the proof as it easily follows by the use of the Eqs. (16), (17), (6), (7), (20), (13)in addition to the fact that $\mathrm{\Pi }$ is a horizontal conformal submersion, using Lemma 2.

Theorem 5

Let $\mathrm{\Pi }$ be a CGS from a l.p.R manifold $(M,g,\mathcal{F})$ onto a RM $(B,h)$ . Then,

1. The distribution $\mathbb{D}$ defines a totally geodesic foliation on M iff $$h\left(\right(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_1,\alpha V_1),{\mathrm{\Pi }}_{\ast }\beta U_2)={\lambda }^{2}g({\widehat{\nabla }}_{U_1}\alpha V_1,\alpha U_2)$$ and $$h\left(\right(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_1,\alpha V_1),{\mathrm{\Pi }}_{\ast }C\mathbb{X})=-{\lambda }^{2}g({\widehat{\nabla }}_{U_1}\alpha B\mathbb{X}+{\mathcal{T}}_{U_1}\beta B\mathbb{X},V_1)\text{.}$$

2. The distribution ${\mathbb{D}}_2$ defines a totally geodesic foliation on M iff $$h\left(\right(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_2,\mathcal{F}{U}_1),{\mathrm{\Pi }}_{\ast }(\beta V_2\left)\right)={\lambda }^{2}g({\widehat{\nabla }}_{U_2}\mathcal{F}{U}_1,\alpha V_2)$$ and $$h\left(\right(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_2,V_2),{\mathrm{\Pi }}_{\ast }(\mathcal{F}C\mathbb{X}\left)\right)={\lambda }^2\left\{g({\mathcal{T}}_{U_2}B\mathbb{X},\beta V_2)-g({\widehat{\nabla }}_{U_2}\alpha V_2,B\mathbb{X})\right\},$$

for any $U_1,V_1\in \mathrm{\Gamma }\left(\mathbb{D}\right),U_2\in \mathrm{\Gamma }\left({\mathbb{D}}_1\right)$ and $\mathbb{X}\in \mathrm{\Gamma }{\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)}^{\perp }$ .

Proof

$\left(i\right)$ For any $U_1,V_1\in \mathrm{\Gamma }\left(\mathbb{D}\right),U_2\in \mathrm{\Gamma }\left({\mathbb{D}}_1\right)$ and $Z\in \mathrm{\Gamma }{\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)}^{\perp }$ , using (16) and (17) and Proposition (2), we have

$$\begin{array}{c}\hfill g({\nabla }_{U_1}{V}_1,U_2)=g({\nabla }_{U_1}\mathcal{F}{V}_1,\alpha U_2)+g({\nabla }_{U_1}\mathcal{F}{V}_1,\beta U_2)\\ \hfill & =g({\widehat{\nabla }}_{U_1}\alpha V_1,\alpha U_2)+g(\mathcal{H}{\nabla }_{U_1}\alpha V_1,\beta U_2)\text{.}\hfill \end{array}$$

Furthermore, using (13), we get

$$g({\nabla }_{U_1}{V}_1,U_2)=g({\widehat{\nabla }}_{U_1}\alpha V_1,\alpha U_2)-{\lambda }^{-2}h\left(\right(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_1,\alpha V_1),{\mathrm{\Pi }}_{\ast }\beta U_2)\text{.}$$

Taking into account that $\mathrm{\Pi }$ is a CGS and using (5), (13), we obtain

$$\begin{array}{cc}\hfill g({\nabla }_{U_1}{V}_1,Z)& =-g(V_1,{\widehat{\nabla }}_{U_1}\alpha \mathit{BZ})-g(V_1,{\mathcal{T}}_{U_1}\beta \mathit{BZ})\hfill \\ \hfill & -{\lambda }^{-2}h\left(\right(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_1,\alpha V_1),{\mathrm{\Pi }}_{\ast }\mathit{CZ})\text{.}\hfill \end{array}$$

Therefore, by the virtue of (28) and (29), we obtain $\left(i\right)$ .

$\left(\mathit{ii}\right)$ For any vector fields $U_1\in \mathrm{\Gamma }\mathbb{D})$ and $U_2,V_2\in \mathrm{\Gamma }\left({\mathbb{D}}_1\right)$ , using (16)(17)

$$\begin{array}{cc}\hfill g({\nabla }_{U_2}{V}_2,U_1)& =g(\mathcal{F}{\nabla }_{U_2}{V}_2,\mathcal{F}{U}_1)\hfill \\ \hfill & =-g({\nabla }_{U_2}\mathcal{F}{U}_1,\mathcal{F}{V}_2)\hfill \\ \hfill & =-g({\widehat{\nabla }}_{U_2}\mathcal{F}{U}_1,\alpha V_2)-g({\widehat{\nabla }}_{U_2}\mathcal{F}{U}_1,\beta V_2)\hfill \\ \hfill & =-g({\widehat{\nabla }}_{U_2}\mathcal{F}{U}_1,\alpha V_2)+g\left(\right(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_2,\mathcal{F}{U}_1),{\mathrm{\Pi }}_{\ast }\beta V_2)\text{.}\hfill \end{array}$$

Moreover, for any $\mathbb{X}\in \mathrm{\Gamma }{\left(\mathit{ker}{\mathrm{\Pi }}_{\ast }\right)}^{\perp }$

$$\begin{array}{cc}\hfill g({\nabla }_{U_2}{V}_2,\mathbb{X})& =g(\mathcal{F}{\nabla }_{U_2}{V}_2,\mathcal{F}\mathbb{X})\hfill \\ \hfill & =g({\nabla }_{U_2}\alpha V_2,B\mathbb{X})+g({\nabla }_{U_2}\beta V_2,B\mathbb{X})+g({\nabla }_{U_2}\mathcal{F}{V}_2,C\mathbb{X})\hfill \\ \hfill & =g({\widehat{\nabla }}_{U_2}\alpha V_2,B\mathbb{X})-g({\mathcal{T}}_{U_2}B\mathbb{X},\beta V_2)-g\left(\right(\nabla {\mathrm{\Pi }}_{\ast }\left)\right(U_2,V_2),{\mathrm{\Pi }}_{\ast }(\mathcal{F}C\mathbb{X}\left)\right)\text{.}\hfill \end{array}$$

Therefore, with (30) and (31), we arrive at $\left(\mathit{ii}\right)$ .