1. Introduction
Einstein Riemannian 4-manifolds
$(M,g)$
with a parallel, isometric, almost paracomplex structure
$P$
exhibit many interesting properties through the metric
$g'$
defined by
$g'=g(P.,.)$
. In particular, the metric
$g'$
is of neutral signature, locally conformally flat, and scalar flat and shares the same Levi-Civita connection and Ricci tensor with
$g$
[Reference Georgiou and Guilfoyle5].
Recently, Urbano in [Reference Urbano11] and later Gao et al. in [Reference Gao, Ma and Yao4] have studied hypersurfaces in
${\mathbb S}^2\times{\mathbb S}^2$
and
${\mathbb H}^2\times{\mathbb H}^2$
, respectively, endowed with the Einstein product metric. In particular, they used two complex structures
$J_1,J_2$
on those manifolds to study isoparametric and homogeneous hypersurfaces by considering the product
$P=J_1J_2$
, which is an (almost) paracomplex structure that is parallel and isometric with respect to the product metric.
The space
${\mathbb L}(M^3)$
of oriented geodesics in the three-dimensional non-flat real space form
$M^3$
is a four-dimensional manifold admiting an Einstein metric and a paracomplex structure
$P$
that is isometric and parallel. Therefore, there exists a neutral, locally conformally flat and scalar flat metric sharing the same Levi-Civita connection and Ricci tensor with the Einstein metric (see [Reference Alekseevsky, Guilfoyle and Klingenberg1] and [Reference Anciaux2] for more details). The paracomplex structure
$P$
has been explicitly described by Anciaux in [Reference Anciaux2] in a similar manner as in the product of surfaces. More precisely, Anciaux constructed two (para) complex structures
$J_1$
and
$J_2$
so that
$J_1J_2=J_2J_1$
and then considered the product
$P=J_1J_2$
. This paracomplex structure was used in [Reference Georgiou and Guilfoyle6], to study a class of hypersurfaces in
${\mathbb L}(M^3)$
, called tangential congruences, that are sets of all tangent-oriented geodesics in a given surface in
$M$
. Particularly, it was shown that tangential congruences are null with respect to the neutral metric and if, additionally, they are tangent to a convex surface then they admit a contact structure. The space
${\mathbb L}({\mathbb R}^3)$
of oriented lines in
${\mathbb R}^3$
is also a four-dimensional manifold admiting a neutral metric
$G$
that is locally conformally flat and scalar flat and is invariant under the Euclidean motions [Reference Alekseevsky, Guilfoyle and Klingenberg1, Reference Guilfoyle and Klingenberg8]. M. Salvai showed that
$G$
is the only metric that is invariant of the group action of the Eucliean 3-space. The null hypersurfaces in
${\mathbb L}({\mathbb R}^3)$
play an important role in the study of the ultrahyperbolic equation:
\begin{equation} u_{x_1x_1}+u_{x_2x_2}-u_{x_3x_3}-u_{x_4x_4}=0, \end{equation}
where
$u=u(x_1,x_2,x_3,x_4)$
is a real function in
${\mathbb R}^4$
(see [Reference Cobos and Guilfoyle3]). Specifically, let
${\mathbb R}^{2,2}=({\mathbb R}^4, g_0\;:\!=\;dx_1^2+dx_2^2-dx_3^2-dx_4^2)$
, and
$f:{\mathbb L}({\mathbb R}^3)\rightarrow{\mathbb R}^{2,2}$
be the conformal map defined according to
$G=\omega ^2f^{\ast }g_0$
, where
$\omega$
is a strictly positive function. A function
$v$
is harmonic with respect to
$G$
, that is,
$\Delta _{G}u=0$
, if and only if
$\omega \cdot v\circ f$
is a solution of the ultrahyperbolic equation (1.1) [Reference Cobos and Guilfoyle3]. This implies solving the ultrahyperbolic equation is equivalent to solving the Laplace equation with respect to the neutral metric
$G$
. Consider now the problem:
\begin{equation*} \Delta _G v=0, \end{equation*}
where the function
$v$
on
${\mathbb L}({\mathbb R}^3)$
is given on the null hypersurface
$H=\{\gamma \in{\mathbb L}({\mathbb R}^3)|\,\, \gamma \parallel P_0\}$
, with
$P_0$
is a fixed plane in
${\mathbb R}^3$
. In [Reference Guilfoyle7], Guilfoyle presented an inversion formula describing
$v$
on
${\mathbb L}({\mathbb R}^3)$
, using Fritz John’s inversion formula (cf. [Reference John9]). It is then natural to ask whether an arbitrary real function defined on a null hypersurface can be uniquely extended to a harmonic function on
${\mathbb L}(M^3)$
with respect to the neutral metric, for any three-dimensional real space form
$M^3$
.
In this article, we study null hypersurfaces with respect to the neutral metric
$g_-$
of an Einstein four-dimensional manifold
$(M,g_+)$
endowed with an almost paracomplex structure
$P$
that is parallel and isometric so that
$g_-=g(P_+.,.)$
.
Our first result deals with totally geodesic null hypersurfaces. In particular, we have the following:
Theorem 1.
Every totally geodesic null hypersurface is scalar flat. If
$M$
admits a totally geodesic null hypersurface then
$(M,g_+)$
is Ricci-flat.
Let
$N$
be the unit normal vector field, with respect to the Riemannian Einstein metric
$g_+$
along a null hypersurface. The principal curvature corresponding to the principal direction
$PN$
is zero. The other two principal curvatures are called nontrivial. The next result provides a necessary condition for the existence of null hypersurfaces with equal nontrivial principal curvatures.
Theorem 2.
Suppose
$(M,g)$
has nonnegative scalar curvature and
$\Sigma$
is a null hypersurface with equal nontrivial principal curvatures. Then,
$g$
is Ricci-flat and
$\Sigma$
is totally geodesic.
Finally, we study (non-minimal) null hypersurfaces having constant mean curvature (CMC). In particular, we prove the following:
Theorem 3.
Let
$\Sigma$
be a CMC, non-minimal null hypersurface in
$(M,g)$
. Then, all principal curvatures and the scalar curvature of
$\Sigma$
are constant. Furthermore, the scalar curvature of
$g$
is given by:
\begin{equation*} \bar R=-8\lambda _1\lambda _2, \end{equation*}
where
$\lambda _1,\lambda _2$
, denote the nontrivial principal curvatures of
$\Sigma$
.
2. Preliminaries
Let
$(M,g)$
be an Einstein 4-manifold endowed with a product structure
$P$
(specifically a type (1,1) tensor field with
$P^2=\mbox{Id}$
) such that:
-
1. The eigenbundles corresponding to the eigenvalues
$+1$
and
$-1$
have equal rank. -
2.
$P$
is an isometry, that is,
\begin{equation*}g(P.,P.)=g(.,.).\end{equation*}
-
3.
$P$
is parallel, that is,where
\begin{equation*}\overline \nabla P=0,\end{equation*}
$\overline \nabla$
is the Levi-Civita connection of
$g$
.
In other words,
$P$
is an almost paracomplex structure that is parallel and isometric.
Define the metric
$g_-$
by:
\begin{equation*}g_-=g(P.,.),\end{equation*}
and denote
$g$
by
$g_+$
. Then,
$g_-$
is of neutral signature, locally conformally flat and scalar flat [Reference Georgiou and Guilfoyle5]. Also, both metrics
$g_+$
and
$g_-$
share the same Levi-Civita connection
$\overline \nabla$
(see [Reference Anciaux2] for further details).
Let
$\Sigma ^3$
be an oriented hypersurface of
$M$
and consider the normal bundles:
\begin{equation*} \mathcal {N}_\pm (\Sigma )=\{\xi \in TM\,|\, g_\pm (X,\xi )=0,\, \forall X\in T\Sigma \}. \end{equation*}
Let
$N_\pm$
be the normal vector of
$\Sigma$
with respect to
$g_\pm$
so that
\begin{equation*} g_\pm (N_\pm,N_\pm )=\epsilon _\pm \in \{-1,0,1\}, \end{equation*}
(note that
$\epsilon _+=1$
) and define the functions
$C_{\pm }$
on
$\Sigma$
according to
\begin{equation*} C_+=g_+(PN_+,N_+)=g_-(N_+,N_+), \end{equation*}
and
\begin{equation*} C_-=g_-(PN_-,N_-)=g_+(N_-,N_-). \end{equation*}
Consider the tangential vector field along
$\Sigma$
:
\begin{equation*} X=PN_+-C_+ N_+. \end{equation*}
Let
$\nabla$
be the Levi-Civita connection of
$g_+$
induced on
$\Sigma$
. For a tangential vector field
$Y$
along
$\Sigma$
, we have
\begin{eqnarray*} g_+(\nabla C_+,Y)&=&\nabla _Y C_+\\[5pt] &=&2\,g_+(\overline \nabla _Y N_+,X)\\[5pt] &=&g_+(Y,-2A_+ X), \end{eqnarray*}
showing that
\begin{equation} \nabla C_+=-2A_+ X, \end{equation}
where
$A_\pm$
denotes the shape operator of
$\Sigma$
immersed in
$(M,g_\pm )$
.
Also,
\begin{equation} \nabla _Y X=-P^T A_+Y+C_+ A_+Y, \end{equation}
where
$P^T$
stands for the orthogonal projection of
$P$
on
$\Sigma$
. Let
$R_\pm, H_\pm$
, and
$\sigma _\pm$
be, respectively, the scalar curvature, the mean curvature, and the second fundamental form of
$\Sigma$
immersed in
$(M,g_\pm )$
.
Proposition 1.
The Hessian of
$C_+$
is
\begin{equation} \nabla ^2C_+(u,v)=-2(\nabla _u\sigma _+)(X,v)-2 C_+ g_+(A_+ u,A_+ v)+2g_+ (PA_+ u,A_+ v). \end{equation}
Proof. In this proof, we omit the subscript
$+$
unless it is necessary.
Using (2.1) on the tangential vector fields
$u,v$
, we have
\begin{eqnarray*} \nabla ^2C(u,v)&=& g(\nabla _u(-2AX),v)\\[5pt] &=& -2g(\nabla _uAX,v)\\[5pt] &=& -2\nabla _u(g(AX,v))+2g(AX,\nabla _uv)\\[5pt] &=& -2\nabla _u(g(X,Av))+2g(AX,\nabla _uv)\\[5pt] &=& -2g(\nabla _uX,Av)-2g(X,\nabla _uAv)+2g(AX,\nabla _uv)\\[5pt] &=& -2g(\epsilon CAu-P^TAu,Av)-2g(X,\nabla _uAv)+2g(AX,\nabla _uv)\\[5pt] &=& -2\epsilon Cg(Au,Av)+2G(PAu,Av)-2g(X,\nabla _uAv)+2g(AX,\nabla _uv) \end{eqnarray*}
Note that
$\sigma (u,v)=g(Au,v)$
and for simplicity use
$\nabla _u\sigma (X,v)$
to denote
$(\nabla _u\sigma )(X,v)$
. We now have
\begin{eqnarray*} \nabla _u\sigma (X,v)&=& u(\sigma (X,v))-\sigma (\nabla _uX,v)-\sigma (X,\nabla _uv)\\[5pt] &=& u(G(X,Av))-g(\nabla _uX,Av)-g(AX,\nabla _uv)\\[5pt] &=& g(\nabla _uX,Av)+g(X,\nabla _uAv)-g(\nabla _uX,Av)-g(AX,\nabla _uv)\\[5pt] &=&g(X,\nabla _uAv)-g(AX,\nabla _uv), \end{eqnarray*}
and therefore,
\begin{align*} \nabla ^2C(u,v)=-2\epsilon Cg(Au,Av)+2g(PAu,Av)-2\nabla _u\sigma (X,v). \end{align*}
Proposition 2.
If
$\Delta$
denotes the Laplacian of the metric
$g_+$
induced on the hypersurface
$\Sigma$
, then
\begin{equation*} \Delta C_+=-6\,g_+(X_+,\nabla H_+)-2C_+|\sigma _+|^2+2\,\textit {Tr}(P^TA_+^2), \end{equation*}
where
$H_+$
denotes the mean curvature and
$A_+$
is the shape operator.
Proof. In the proof, we omit the subscript
$+$
unless it is necessary. The Codazzi–Mainardi equation for
$\Sigma$
is
\begin{equation*} g(R(u,v)z,N)=(\nabla _u\sigma )(v,z)-(\nabla _v\sigma )(u,z). \end{equation*}
Consider the orthonormal frame
$(e_1,e_2,e_3)$
of
$\Sigma$
, where
$Ae_i=\lambda _i e_i$
. The fact that
$g$
is Einstein gives
\begin{eqnarray*} \sum _{i=1}^3\left ((\nabla _{e_i}\sigma )(X,e_i)-(\nabla _X\sigma )(e_i,e_i)\right )&=&\sum _{i=1}^3g(R(e_i,X)e_i,N)\\[5pt] &=&\sum _{i=1}^3g(R(e_i,X)e_i,N)+g(R(N,X)N,N)\\[5pt] &=&\overline{\mbox{Ric}}(X,N)\\[5pt] &=&\textstyle{\frac{\bar R}{4}}\,g(X,N)\\[5pt] &=&0. \end{eqnarray*}
Thus,
\begin{eqnarray*} \sum _{i=1}^3(\nabla _{e_i}\sigma )(X,e_i)&=&\sum _{i=1}^3(\nabla _X\sigma )(e_i,e_i)\\[5pt] &=&\sum _{i=1}^3\nabla _X(\sigma (e_i,e_i))-\sigma (\nabla _Xe_i,e_i)-\sigma (e_i,\nabla _Xe_i)\\[5pt] &=&3\nabla _XH-2\sum _{i=1}^3g(\nabla _Xe_i,Ae_i)\\[5pt] &=&3g(X,\nabla H)-2\sum _{i=1}^3\lambda _i g(\nabla _Xe_i,e_i)\\[5pt] &=&3g(X,\nabla H). \end{eqnarray*}
We now have
\begin{eqnarray*} \Delta C&=&\sum _{i=1}^3\nabla ^2C(e_i,e_i)\\[5pt] &=&-2\sum _{i=1}^3\left ((\nabla _{e_i}\sigma )(X,e_i)+ Cg(Ae_i,Ae_i)-g(PAe_i,Ae_i)\right ) \\[5pt] &=&-6g(X,\nabla H)-2\sum _{i=1}^3\left (\lambda ^2_iC-\lambda ^2_ig(Pe_i,e_i)\right ), \end{eqnarray*}
and this completes the proof.
Let
$R,R_{ij}, R_{ijkl}$
be, respectively, the scalar curvature, the Ricci tensor, and the curvature tensor of the metric
$g_+$
induced on
$\Sigma$
and let
$\bar R,\bar R_{ij}, \bar R_{ijkl}$
be, respectively, the scalar curvature, the Ricci tensor, and the curvature of the ambient metric
$g_+$
.
Using the Gauss equation, we get (for simplicity, we omit the subscript
$+$
):
\begin{eqnarray*} R&=& g^{ij}R_{ij}\\[5pt] &=& g^{ij}g^{kl}(\bar R_{kilj}+\sigma _{ij}\sigma _{kl}-\sigma _{il}\sigma _{kj})\\[5pt] &=& g^{ij}g^{kl}\bar R_{kilj}+9H^2-|\sigma |^2. \end{eqnarray*}
The fact the
$g_+$
is Einstein implies
\begin{eqnarray*} g^{ij}g^{kl}\bar R_{kilj}&=& g^{ij}\bar R_{ij} -g^{NN}\overline{\mbox{Ric}}(NN) \\[5pt] &=&(\bar R_+-g^{NN}\overline{\mbox{Ric}}(NN))-g^{NN}\overline{\mbox{Ric}}(NN)\\[5pt] &=&\bar R_+-2\overline{\mbox{Ric}}(NN)\\[5pt] &=&\bar R_+-2(\bar R_+/4)g_+(N,N)\\[5pt] &=&\bar R_+/2. \end{eqnarray*}
We then have
\begin{equation} R_+=\textstyle{\frac{1}{2}}\bar R_++9H_+^2-|\sigma _+|^2. \end{equation}
We then have
Proposition 3.
Assume
$(M,g_+)$
has positive (resp. negative) scalar curvature and
$\Sigma$
is a totally geodesic hypersurface. Then the metric
$g_+$
induced on
$\Sigma$
has positive (resp. negative) scalar curvature.
3. Null hypersurfaces
Definition 1. A null hypersurface in a pseudo-Riemannian manifold is an oriented hypersurface where the induced metric is indefinite and the normal vector field is null.
In this section, when we refer to a null hypersurface we simply mean a hypersurface that is null with respect to the neutral metric of
$g_-$
.
Proposition 4.
Suppose
$\Sigma$
is an oriented hypersurface of
$M$
. Then, the following statements hold:
-
1.
$|C_+|\leq 1,\quad \textit{and}\quad C_-\gt 0$
. -
2.
$C_+=0$
, if and only if
$\Sigma$
is a null hypersurface.
-
3. If
$\Sigma$
is a null hypersurface, then
$PN_+$
is a principal direction with zero corresponding principal curvature.
Proof.
-
1. It is not hard to confirm that
$|X|^2=1-(C_+)^2\geq 0$
. Also,
\begin{equation*} C_-=g_+(N_-,N_-)\gt 0. \end{equation*}
-
2. Assuming
$C_+=0$
, we have that
$g_+(PN_+,N_+)=0$
and using the fact that
$g_+$
is Riemannian then,
$PN_+\in T\Sigma$
. This impliesor,
\begin{equation*} g_-(PN_+,N_-)=0, \end{equation*}
But this tells us that
\begin{equation*} g_+(N_+,N_-)=0. \end{equation*}
$N_-\in T\Sigma$
, and thereforewhich means that
\begin{equation*} g_-(N_-,N_-)=0, \end{equation*}
$\Sigma$
is null. Conversely, assume that
$\Sigma$
is null and consider the nonzero normal vector field
$N_-$
. Then,
$g_-(N_-,N_-)=0$
. On the other hand,
$g_-(N_-,T\Sigma )=0$
, which means
$g_+(PN_-,T\Sigma )=0$
. Therefore,
$PN_-=\lambda N_+$
, where
$\lambda \neq 0$
, since
$N_-$
is nonzero vector field. Thus,and this completes the proof.
\begin{eqnarray*} C_+&=& g_-(N_+,N_+)\\[5pt] &=& \lambda ^{-2}g_-(N_-,N_-)\\[5pt] &=&0, \end{eqnarray*}
-
3. Since
$\Sigma$
is null then
$C_+=0$
and therefore,Note that
\begin{equation*} X_+=PN_+-C_+N_+=PN_+\in T\Sigma. \end{equation*}
which implies
\begin{equation*} 0=\nabla C_+=-2A_+X_+, \end{equation*}
and therefore
\begin{equation*} A_+PN_+=0, \end{equation*}
$PN_+$
is a principal direction.
For a null hypersurface
$\Sigma$
, we study the geometric properties of the metric
$g_+$
induced on
$\Sigma$
and for this reason we omit the
$+$
subscripts unless it is necessary.
3.1. Examples of null hypersurfaces
Example 3.1. We now describe the almost paracomplex structure defined in the spaces of oriented geodesics of 3-manifolds of constant curvature using their (para) Kähler structures (see [Reference Alekseevsky, Guilfoyle and Klingenberg1, Reference Georgiou and Guilfoyle6, Reference Guilfoyle and Klingenberg8, Reference Salvai10] for more details).
For
$p\in \{0,1,2,3\}$
, consider the (pseudo-) Euclidean 4-space
${\mathbb R}_p^4\;:\!=\;({\mathbb R}^4,\left \lt .,.\right \gt _p)$
, where
\begin{equation*} \left \lt .,.\right \gt _p=-\sum _{i=1}^p dX_i^2+\sum _{i=p+1}^4dX_i^2, \end{equation*}
and let
${\mathbb S}_p^{3}$
be the quadric
\begin{equation*} {\mathbb S}_p^{3}=\{x\in {\mathbb R}^4|\; \left \lt x,x\right \gt _p=1\}. \end{equation*}
The quadric
${\mathbb S}_0^{3}$
is the 3-sphere
${\mathbb S}^{3}$
,
${\mathbb S}_3^{3}\cap \{x\in{\mathbb R}^4|\, X_4\gt 0\}$
is anti-isometric to the hyperbolic 3-space
${\mathbb H}^{3}$
,
${\mathbb S}_1^{3}$
is the de Sitter 3-space
$d{\mathbb S}^{3}$
, and
${\mathbb S}_2^{3}$
is anti-isometric to the anti-de Sitter 3-space
$Ad{\mathbb S}^{3}$
.
Let
$g_p$
be the metric
$\left \lt .,.\right \gt _p$
induced on
${\mathbb S}_p^{3}$
by the inclusion map. The space of oriented geodesics in
${\mathbb S}_p^{3}$
is a four-dimensional manifold and is identified with the following Grasmmannian spaces of oriented planes on
${\mathbb R}_p^4$
:
\begin{equation*} {\mathbb L}^{\pm }({\mathbb S}_p^{3})=\{x\wedge y\in \Lambda ^2({\mathbb R}_p^4)|\; y\in T_x{\mathbb S}_p^{3},\; g_p(y,y)=\pm 1\}. \end{equation*}
Let
$\iota \;:\; {\mathbb L}^{\pm }({\mathbb S}_p^{3})\rightarrow \Lambda ^2({\mathbb R}_p^4)$
be the inclusion map and
$\left \lt \left \lt,\right \gt \right \gt _p$
be the flat metric in the 6-manifold
$\Lambda ^2({\mathbb R}_p^4)$
defined by:
\begin{equation*} \left \lt \left \lt u_1\wedge v_1,u_2\wedge v_2\right \gt \right \gt _p\;:\!=\;\left \lt u_1,u_2\right \gt _p\left \lt v_1,v_2\right \gt _p-\left \lt u_1,v_2\right \gt _p\left \lt u_2,v_1\right \gt _p. \end{equation*}
The metric
$G_p=\iota ^{\ast }\left \lt \left \lt,\right \gt \right \gt _p$
on
${\mathbb L}^{\pm }({\mathbb S}_p^{3})$
is Einstein [Reference Anciaux2].
It was shown in [Reference Georgiou and Guilfoyle5], that the Hodge star operator
$\ast$
on the space of bivectors
$\Lambda ^2({\mathbb R}_p^4)$
in
${\mathbb R}_p^4$
, restricted to the space of oriented geodesics
${\mathbb L}^{\pm }({\mathbb S}^3_p)$
defines an almost paracomplex structure
${\mathbb J}^{\ast }$
that is parallel and isometric with respect to the Einstein metric
$G_p$
. In particular, for
$x\wedge y\in{\mathbb L}^{\pm }({\mathbb S}_p^{3})$
, the almost paracomplex structure is defined by:
\begin{equation*} {\mathbb J}^{\ast }_{x\wedge y}=\left. \ast \right |_{T_{x\wedge y}{\mathbb L}^{\pm }({\mathbb S}^3_p)}. \end{equation*}
The metric
$G'_p\;:\!=\;G_p({\mathbb J}^{\ast } .,.)$
, is of neutral signature, locally conformally flat and scalar flat in
${\mathbb L}^{\pm }({\mathbb S}_p^{3})$
.
Let
$\phi \;:\; S\rightarrow{\mathbb S}_p^3$
be a non-totally geodesic smooth surface and
$(e_1,e_2)$
be the principal directions of
$\phi$
with corresponding eigenvalues
$\kappa _1$
and
$\kappa _2$
. Then,
\begin{equation*} \Phi \;:\; S\times {\mathbb S}^1\rightarrow {\mathbb L}({\mathbb S}_p^3)\;:\;(x,\theta )\mapsto \phi (x)\wedge (\cos \theta \, e_1(x)+\sin \theta \, e_2(x)), \end{equation*}
is the immersion of the tangential congruence
$\Sigma =\Phi (S\times{\mathbb S}^1)$
in the space of oriented geodesics
${\mathbb L}({\mathbb S}^3_p)$
. It can be shown that if
$\phi$
is a totally geodesic immersion, the mapping
$\Phi$
is not an immersion. Also,
$\Sigma$
is a null hypersurface with respect to the locally conformally flat neutral metric
$g_-$
[Reference Georgiou and Guilfoyle6].
The eigenvalues of the tangential hypersurface
$\Sigma$
are
$0,\lambda _+$
and
$\lambda _-$
, where
\begin{equation*} \lambda _+=\kappa _1\cos ^2\theta +\kappa _2\sin ^2\theta \qquad \lambda _-=-\kappa _1\sin ^2\theta -\kappa _2\cos ^2\theta, \end{equation*}
and therefore the mean curvature is
\begin{equation*} H= \textstyle{\frac {1}{3}}(\kappa _1-\kappa _2)\cos 2\theta. \end{equation*}
This yields
Proposition 5.
If
$S$
is a totally umbilic surface in the non-flat three-dimensional real space form, then the corresponding tangential congruence
$\Sigma$
is a null hypersurface in
$({\mathbb L}({\mathbb S}^3_p),G'_p)$
and is minimal in
$({\mathbb L}({\mathbb S}^3_p),G_p)$
.
Example 3.2.
Consider the Cartesian product of the 2-spheres
${\mathbb S}^2\times{\mathbb S}^2$
endowed with the product metric:
\begin{equation*} g_+=g\oplus g, \end{equation*}
where
$g$
is the round metric of
${\mathbb S}^2$
. It is well known that
$g_+$
is Einstein with scalar curvature
$R=4$
.
Define the almost paracomplex structure
$P$
on
${\mathbb S}^2\times{\mathbb S}^2$
by:
\begin{equation*} P(u,v)=(u,-v), \end{equation*}
where
$(u,v)\in T({\mathbb S}^2\times{\mathbb S}^2)$
. Then,
$P$
is
$G^+$
-parallel and isometric. For
$t\in (-1,1)$
, consider the homogeneous hypersurfaces:
\begin{equation*} \Sigma _t=\{(x,y)\in {\mathbb S}^2\times {\mathbb S}^2\subset {\mathbb R}^3\times {\mathbb R}^3)\,|\, \left \lt x,y\right \gt =t\}. \end{equation*}
In fact,
$\Sigma _t$
is a tube of radius
$\cos ^{-1}(t/\sqrt{2})$
over the diagonal surface
$\Delta =\{(x,x)\in{\mathbb S}^2\times{\mathbb S}^2\}$
. It was shown in [Reference Urbano11] that
$\Sigma _t$
is null for every
$t$
with respect to the neutral metric:
\begin{equation*} g_-=g_+(P.,.)=g\oplus (-g) \end{equation*}
and the principal curvatures are
\begin{equation*} \lambda _1= \textstyle{\frac {1}{\sqrt {2}}}\sqrt {\frac {1+t}{1-t}},\qquad \lambda _2=-\frac {1}{\sqrt {2}}\sqrt {\frac {1-t}{1+t}},\qquad \lambda _3=0. \end{equation*}
Thus,
$\Sigma _t$
is a CMC null hypersurface for any
$t\in (-1,1)$
and is minimal only when
$t=0$
as the mean curvature
$H$
is
\begin{equation*} H= \textstyle{\frac {1}{3\sqrt {2}}}\left (\sqrt {\frac {1+t}{1-t}}-\sqrt {\frac {1-t}{1+t}}\right ). \end{equation*}
Similarly, we have the following example.
Example 3.3.
Consider the Cartesian product of the 2-spheres
${\mathbb H}^2\times{\mathbb H}^2$
endowed with the product metric:
\begin{equation*} g_+=g\oplus g, \end{equation*}
where
$g$
is the standard hyperbolic metric of
${\mathbb H}^2$
. It is not hard for one to see that
$g_+$
is Einstein with scalar curvature
$R=-4$
. As before, the almost paracomplex structure
$P$
on
${\mathbb H}^2\times{\mathbb H}^2$
is given by:
\begin{equation*} P(u,v)=(u,-v), \end{equation*}
where
$(u,v)\in T({\mathbb H}^2\times{\mathbb H}^2)$
. Again,
$P$
is
$g_+$
-parallel and isometric and for
$t\in (-\infty,-1)$
, consider the homogeneous hypersurfaces:
\begin{equation*} \Sigma _t=\{(x,y)\in {\mathbb H}^2\times {\mathbb H}^2\subset {\mathbb R}^3\times {\mathbb R}^3)\,|\, \left \lt x,y\right \gt =t\}. \end{equation*}
In fact,
$\Sigma _t$
is a tube of radius
$\cosh ^{-1}(t/\sqrt{2})$
over the diagonal surface
$\Delta =\{(x,x)\in{\mathbb H}^2\times{\mathbb H}^2\}$
. It was shown in [Reference Gao, Ma and Yao4] that
$\Sigma _t$
is null for every
$t$
with respect to the neutral metric:
\begin{equation*} g_-=g_+(P.,.)=g\oplus (-g) \end{equation*}
and the principal curvatures are
\begin{equation*} \lambda _1= \textstyle{\frac {1}{\sqrt {2}}}\sqrt {\frac {1+t}{1-t}},\qquad \lambda _2=\frac {1}{\sqrt {2}}\sqrt {\frac {1-t}{1+t}},\qquad \lambda _3=0. \end{equation*}
Thus,
$\Sigma _t$
is a CMC, non-minimal null hypersurface for any
$t\in (-1,1)$
with mean curvature:
\begin{equation*} H= \textstyle{\frac {1}{3\sqrt {2}}}\left (\sqrt {\frac {1+t}{1-t}}+\sqrt {\frac {1-t}{1+t}}\right ). \end{equation*}
3.2. Main results
Consider the principal orthonormal frame
$(e_1,e_2,e_3=PN)$
of the null hypersurface
$\Sigma$
so that
\begin{equation*} Ae_i=\lambda _i e_i. \end{equation*}
It is easily shown that there is an angle
$\theta \in [0,2\pi )$
such that
\begin{equation*} Pe_1=\cos \theta e_1+\sin \theta e_2\qquad Pe_2=\sin \theta e_1-\cos \theta e_2. \end{equation*}
We call the angle
$\theta$
the principal angle of the null hypersurface
$\Sigma$
.
We now have the following result for totally geodesic null hypersurfaces:
Theorem 1.
Every totally geodesic null hypersurface is scalar flat. If
$M$
admits a totally geodesic null hypersurface, then
$(M,g_+)$
is Ricci-flat.
Proof. Let
$\{e_1,e_2.e_3\}$
be an orthonormal frame of
$\Sigma$
such that
\begin{equation*} Ae_i=\lambda _i e_i, \end{equation*}
where
$e_3=PN$
and therefore,
$\lambda _3=0$
. The almost paracomplex structure
$P$
is
\begin{equation*} P=\begin {pmatrix}\cos \theta\;\;\;\;\; & \sin \theta\;\;\;\;\; & 0\;\;\;\;\; & 0\\[5pt] \sin \theta\;\;\;\;\; & -\cos \theta\;\;\;\;\; & 0\;\;\;\;\; & 0\\[5pt] 0\;\;\;\;\; & 0\;\;\;\;\; & 0\;\;\;\;\; & 1\\[5pt] 0\;\;\;\;\; & 0\;\;\;\;\; &1\;\;\;\;\; & 0 \end {pmatrix}. \end{equation*}
with respect to the orthonormal frame
$(e_1,e_2,e_3,N)$
.
Let
$\overline \nabla,\nabla$
be the Levi-Civita connections for the metrics
$g$
and the induced metric of
$g$
on
$\Sigma$
, respectively. For
$i,j=1,2,3$
, we have
\begin{equation*} \overline \nabla _{e_i}e_j=\nabla _{e_i}e_j+\lambda _i\delta _{ij}N, \end{equation*}
and if we let
$\omega _{ij}^k=g(\nabla _{e_i}e_j,e_k)$
then
\begin{equation*} \omega _{ij}^k=-\omega _{ik}^j. \end{equation*}
Defining
\begin{equation} k=\omega _{11}^2,\qquad \mu =\omega _{21}^2,\qquad \nu =\omega _{31}^2. \end{equation}
A brief calculation gives
\begin{equation*} g(R(e_2,e_1)e_1,e_2)=-e_1(\mu )+e_2(k)+\lambda _1\lambda _2-k^2-\mu ^2+\nu (\lambda _1-\lambda _2)\sin \theta. \end{equation*}
\begin{equation*} g(R(e_3,e_1)e_1,e_3)=-\lambda _1\nu \sin \theta -\lambda _2\nu \sin \theta +e_3(\lambda _1\cos \theta )-\lambda _1^2\cos ^2\theta -\lambda _1\lambda _2\sin ^2\theta. \end{equation*}
\begin{equation*} g(R(e_3,e_2)e_2,e_3)=\lambda _1\nu \sin \theta +\lambda _2\nu \sin \theta -e_3(\lambda _2\cos \theta )-\lambda _2^2\cos ^2\theta -\lambda _1\lambda _2\sin ^2\theta. \end{equation*}
Therefore, we deduce
\begin{equation*} \mbox {Ric}(e_1,e_1)=-e_1(\mu )+e_2(k)+e_3(\lambda _1\cos \theta )+\lambda _1\lambda _2\cos ^2\theta -\lambda ^2_1\cos ^2\theta -k^2-\mu ^2-2\nu \lambda _2\sin \theta. \end{equation*}
\begin{equation*} \mbox {Ric}(e_2,e_2)=-e_1(\mu )+e_2(k)-e_3(\lambda _2\cos \theta )+\lambda _1\lambda _2\cos ^2\theta -\lambda ^2_2\cos ^2\theta -k^2-\mu ^2+2\nu \lambda _1\sin \theta. \end{equation*}
\begin{equation*} \mbox {Ric}(e_3,e_3)=e_3[(\lambda _1-\lambda _2)\cos \theta ]-2\lambda _1\lambda _2\sin ^2\theta -(\lambda ^2_1+\lambda ^2_2)\cos ^2\theta. \end{equation*}
The scalar curvature
$R$
of
$\Sigma$
is
\begin{align} R&=-2e_1(\mu )+2e_2(k)+2\lambda _1\lambda _2\cos 2\theta -2(\lambda ^2_1+\lambda ^2_2)\cos ^2\theta \\[5pt] &\quad +2e_3[(\lambda _1-\lambda _2)\cos \theta ] -2k^2-2\mu ^2+2\nu (\lambda _1-\lambda _2)\sin \theta .\nonumber \end{align}
Using the fact that
$P$
is parallel, namely
\begin{equation*} P\overline \nabla _{e_i}e_j=\overline \nabla _{e_i}Pe_j, \end{equation*}
we have
\begin{equation} \omega _{12}^3=\lambda _1\sin \theta,\qquad \omega _{11}^3=\lambda _1\cos \theta, \qquad \omega _{12}^1=e_1(\theta/2), \end{equation}
\begin{equation*} \omega _{21}^3=\lambda _2\sin \theta,\qquad \omega _{22}^3=-\lambda _2\cos \theta, \qquad \omega _{22}^1=e_2(\theta/2), \end{equation*}
\begin{equation*} \omega _{13}^1=-\lambda _1\cos \theta \qquad \omega _{13}^2=-\lambda _1\sin \theta \qquad \omega _{23}^1=-\lambda _2\sin \theta, \end{equation*}
\begin{equation*} \omega _{31}^2=-e_3(\theta/2),\qquad \omega _{31}^3=\omega _{32}^3=0, \end{equation*}
and thus,
\begin{equation*} \nabla _{e_1}e_1=-e_1(\theta/2)e_2+\lambda _1\cos \theta e_3 \qquad \nabla _{e_1}e_2=e_1(\theta/2)e_1+\lambda _1\sin \theta e_3 \end{equation*}
\begin{equation*} \nabla _{e_1}e_3=-\lambda _1\cos \theta e_1-\lambda _1\sin \theta e_2 \qquad \nabla _{e_2}e_1=-e_2(\theta/2)e_2+\lambda _2\sin \theta e_3 \end{equation*}
\begin{equation*} \nabla _{e_2}e_2=e_2(\theta/2)e_1-\lambda _2\cos \theta e_3 \qquad \nabla _{e_2}e_3=-\lambda _2\sin \theta e_1+\lambda _2\cos \theta e_2 \end{equation*}
\begin{equation*} \nabla _{e_3}e_1=-e_3(\theta/2)e_2 \qquad \nabla _{e_3}e_2=e_3(\theta/2)e_1 \qquad \nabla _{e_3}e_3=0. \end{equation*}
The relations (3.1) and (3.3) yield
\begin{equation*} \mu =-e_2(\theta/2),\qquad k=-e_1(\theta/2), \end{equation*}
and therefore,
\begin{equation*} -e_1(\mu )+e_2(k)=[e_1,e_2](\theta/2). \end{equation*}
On the other hand,
\begin{eqnarray*} [e_1,e_2]&=&e_1(\theta/2)e_1+\lambda _1\sin \theta e_3-(-e_2(\theta/2)e_2+\lambda _2\sin \theta e_3)\\[5pt] &=&e_1(\theta/2)e_1+e_2(\theta/2)e_2+(\lambda _1-\lambda _2)\sin \theta \,e_3. \end{eqnarray*}
Thus,
\begin{eqnarray*} -e_1(\mu )+e_2(k)&=&[e_1,e_2](\theta/2)\\[5pt] &=&e_1(\theta/2)e_1(\theta/2)+e_2(\theta/2)e_2(\theta/2)+(\lambda _1-\lambda _2)\sin \theta \,e_3(\theta/2)\\[5pt] &=&k^2+\mu ^2-\nu (\lambda _1-\lambda _2)\sin \theta \end{eqnarray*}
The scalar curvature given in (3.2) now becomes
\begin{equation} R=2e_3[(\lambda _1-\lambda _2)\cos \theta ]+2\lambda _1\lambda _2\cos 2\theta -2(\lambda ^2_1+\lambda ^2_2)\cos ^2\theta. \end{equation}
Assuming that
$\Sigma$
is totally geodesic, we can see easily that
$R=0$
. In this case, the Gauss equation implies also that
$(M,g)$
is scalar flat since
\begin{equation*} \frac {\bar R}{2}=R-9H^2+|\sigma |^2=0. \end{equation*}
The Ricci flatness of
$(M,g)$
follows from the fact
$g$
is Einstein.
If
$\Sigma$
is a null hypersurface, the principal curvature corresponding to the principal direction
$PN$
will be called trivial. The following theorem explores null hypersurfaces where the nontrivial eigenvalues are equal.
Theorem 2.
Suppose
$(M,g)$
has nonnegative scalar curvature and
$\Sigma$
is a null hypersurface with equal nontrivial principal curvatures. Then,
$g$
is Ricci-flat and
$\Sigma$
is totally geodesic.
Proof. Using the scalar curvature
$R$
in (3.4), the Gauss equation for
$\Sigma$
becomes
\begin{equation*} \frac {\bar R}{2}+(\lambda _1+\lambda _2)^2=-(\lambda _1-\lambda _2)^2\cos 2\theta. \end{equation*}
Since
$\lambda _1=\lambda _2$
, we have
\begin{equation*} \frac {\bar R}{2}+(\lambda _1+\lambda _2)^2=0,\end{equation*}
implying
$\bar R=0$
and
$\lambda _1+\lambda _2=0$
. This means that
$\lambda _1=\lambda _2=0$
and thus,
$\Sigma$
is totally null.
We now have the following theorem about CMC null hypersurfaces:
Theorem 3.
Let
$\Sigma$
be a CMC, non-minimal null hypersurface in
$(M,g)$
. Then, all principal curvatures and the scalar curvature of
$\Sigma$
are constant. Furthermore, the scalar curvature of
$g$
is given by:
\begin{equation} \bar R=-8\lambda _1\lambda _2, \end{equation}
where
$\lambda _1,\lambda _2$
, denote the nontrivial principal curvatures of
$\Sigma$
.
Proof. We recall the principal orthonormal frame
$\{e_1,e_2.e_3=PN\}$
of the null hypersurface
$\Sigma$
. The Laplacian of the function
$C$
with respect to the induced metric is
\begin{equation*} \Delta C=-6\,g(X,\nabla H)-2C|\sigma |^2+2\mbox {Tr}(P^TA^2). \end{equation*}
Since
$C=0$
and
$\nabla H=0$
, we have
\begin{equation*} \mbox {Tr}(P^TA^2)=0, \end{equation*}
which ensures
\begin{equation*} \sum _{i=1}^3g(PA^2e_i,e_i)=0. \end{equation*}
It follows
\begin{equation*} \sum _{i=1}^2\lambda ^2_ig(Pe_i,e_i)=0, \end{equation*}
and therefore,
\begin{equation*} (\lambda ^2_1-\lambda ^2_2)\cos \theta =0. \end{equation*}
Note that
$\Sigma$
is non-minimal and therefore,
$\lambda _1+\lambda _2\neq 0$
.
If
$\lambda _1=\lambda _2$
, we have that
$H=\textstyle{\frac{2}{3}}\lambda _1$
is constant and considering the scalar curvature in (3.4), we find
\begin{eqnarray*} \textstyle{\frac{1}{2}}R&=&\lambda ^2_1\cos 2\theta -2\lambda ^2_1\cos ^2\theta \\[5pt] &=&-\lambda ^2_1. \end{eqnarray*}
Using the Gauss equation (2.4), we obtain
\begin{eqnarray*} -2\lambda ^2_1&=& R\\[5pt] &=&\textstyle{\frac{1}{2}}\bar R+9H^2-|\sigma |^2\\[5pt] &=&\textstyle{\frac{1}{2}}\bar R+(2\lambda _1)^2-2\lambda ^2_1, \end{eqnarray*}
which implies that
$\bar R=-8\lambda ^2_1$
.
If
$\cos \theta =0$
, then either
$\theta =\pi/2$
or
$\theta =3\pi/2$
. The scalar curvature of
$\Sigma$
given (3.4) becomes
\begin{equation*} R=-2\lambda _1\lambda _2. \end{equation*}
On the other hand, the scalar curvature in (2.4) yields
\begin{eqnarray*} -2\lambda _1\lambda _2&=& R\\[5pt] &=&\textstyle{\frac{1}{2}}\bar R+9H^2-|\sigma |^2\\[5pt] &=&\textstyle{\frac{1}{2}}\bar R+(\lambda _1+\lambda _2)^2-\lambda ^2_1-\lambda ^2_2, \end{eqnarray*}
and therefore,
$\bar R=-8\lambda _1\lambda _2$
. Note that
$\bar R$
is constant and as such
$\lambda _1\lambda _2$
is constant. However,
$\lambda _1+\lambda _2$
is also constant and thus both
$\lambda _1$
and
$\lambda _2$
are constant.
All principal curvatures are constant, and therefore the Gauss equation, given in (2.4), tells us that the scalar curvature
$R$
must also be constant.
Theorem 5 can no longer be extended to minimal null hypersurfaces, since the relation (3.5) does not necessarily hold. To see this, consider the minimal, null hypersurfaces
$M_{a,b}\subset{\mathbb S}^2\times{\mathbb S}^2$
, for
$a,b\in{\mathbb S}^2\subset{\mathbb R}^3$
:
\begin{equation*} M_{a,b}=\{(x,y)\in {\mathbb S}^2\times {\mathbb S}^2\,|\, \left \lt x,a\right \gt +\left \lt y,b\right \gt =0\}. \end{equation*}
In [Reference Urbano11], Urbano showed that the principal curvatures are nonconstant and in particular, if
$(x,y)\in M_{a,b}$
then:
\begin{equation*} \textstyle{\lambda _1(x,y)}= \textstyle{\frac {\left \lt x,a\right \gt }{\sqrt {2(1-\left \lt x,a\right \gt ^2)}}},\qquad \lambda _2(x,y)=-\frac {\left \lt x,a\right \gt }{\sqrt {2(1-\left \lt x,a\right \gt ^2)}},\qquad \lambda _3(x,y)=0. \end{equation*}
As such
\begin{equation*} -8\lambda _1\lambda _2= \textstyle{\frac {4\left \lt x,a\right \gt ^2}{1-\left \lt x,a\right \gt ^2}}\neq 4=\bar R. \end{equation*}
Acknowledgments
The author would like to thank T. Lyons for his helpful and valuable suggestions and comments.
