$\tilde{J}$-Tangent Affine Hypersurfaces with an Induced Almost Paracontact Structure

Abstract

The subjects of our study are affine hypersurfaces $f:M\to {\mathbb{R}}^{2n+2}$ considered with a transversal vector field C, which is $\tilde{J}$-tangent. By $\tilde{J}$ we understand the canonical paracomplex structure on ${\mathbb{R}}^{2n+2}$. The vector field C induces on the hypersurface f an almost paracontact structure $(\phi ,\xi ,\eta )$. We obtain a complete classification of hypersurfaces admitting a metric induced almost paracontact structure with respect to the second fundamental form. We show that, in this case, the $\tilde{J}$-tangent transversal vector field is restricted to centroaffine and the hypersurface must be a piece of hyperquadric. It is demonstrated that these hyperquadrics have a very specific form. A three-dimensional example is also given. Moreover, we establish an equivalence relation between almost paracontact metric structures, para $\alpha$-contact metric structures, and para $\alpha$-Sasakian structures. Methods of affine differential geometry, as well as paracomplex/paracontact geometry, are used.

1. Introduction

The notion of paracontact metric structures was first proposed by S. Kaneyuki and F. L. Williams in [1]. The significance of paracontact geometry, and in particular para-Sasakian geometry, has been highlighted in recent years by many authors. Its role in pseudo-Riemannian geometry, as well as in mathematical physics, was emphasized in several papers. For example, D. V. Alekseevsky, V. Cortés, A. S. Galaev, and T. Leistner in [2] showed the one-to-one correspondence between para-Sasakian structures and para-Kähler structures on cones of pseudo-Riemannian manifolds. In [3], the notion of an almost para-CR structure was introduced. Such structures are connected in a natural way with almost paracontact structures; more precisely, every almost paracontact structure $(\phi ,\xi ,\eta )$ induces an almost para-CR structure on the distribution $\mathcal{D}=ker\eta$. In [4], V. Cortés, C. Mayer, T. Mohaupt, and F. Saueressing studied affine special para-Kähler geometry, a paracomplex analogue of affine special Kähler geometry, which plays a central role in supersymmetric theories in physics. An interesting result related to para-Sasakian structures was shown by S. Zamkovoy in [5], where he defined a canonical paracontact connection on a paracontact metric manifold and showed that the torsion of this connection vanishes exactly when the structure is para-Sasakian. Finally, in [6], I. Küpeli Erken studied normal almost paracontact metric manifolds, provided they satisfied some additional projective flatness conditions.

Relations between affine differential geometry and paracomplex geometry can be found in [7], where the authors demonstrated that any special para-Kähler manifold is intrinsically an improper affine hypersphere. On the other hand, in [8], the notion of paracomplex affine immersion was introduced, and existence and uniqueness theorems established. It is worth mentioning that affine immersions with an almost product structure have also been studied (see, e.g., [9]).

In [10], the author studied affine hypersurfaces with a J-tangent transversal vector field, where J was the canonical complex structure on ${\mathbb{R}}^{2n+2}\cong {\mathbb{C}}^{n+1}$. It was proved that if the induced almost contact structure is metric relative to the second fundamental form, then it is a Sasakian structure and the hypersurface itself is a piece of hyperquadric. In this paper, we study affine hypersurfaces $f:M\to {\mathbb{R}}^{2n+2}$ with an arbitrary $\tilde{J}$-tangent transversal vector field, where $\tilde{J}$ is the canonical paracomplex structure on ${\mathbb{R}}^{2n+2}$. Such a vector field induces, in a natural way, an almost paracontact structure $(\phi ,\xi ,\eta )$, as well as the second fundamental form h. We prove that if $(\phi ,\xi ,\eta ,h)$ is an almost paracontact metric structure, then it is a para $\alpha$-Sasakian structure with $\alpha =-1$. Moreover, the hypersurface is a piece of a hyperquadric.

Section 2 provides a brief overview of fundamental formulas from affine differential geometry and introduces the concepts of a $\tilde{J}$-tangent transversal vector field and a $\tilde{J}$-invariant distribution $\mathcal{D}$.

In Section 3, we recall the definitions of an almost paracontact metric structure, para $\alpha$-Sasakian structure, and para $\alpha$-contact structure. We introduce the notion of an induced almost paracontact structure and prove some results related to this structure.

Section 4 contains the main results of this paper. We prove that if $(\phi ,\xi ,\eta ,h)$ is an almost paracontact metric structure, then the hypersurface is equiaffine and the shape operator $S=-id$. In consequence, the structure is para $(-1)$-Sasakian. We also prove that the hypersurface is a piece of a hyperquadric and give an explicit formula for it.

2. Preliminaries

In this section, we provide a short summary of the essential formulas of affine differential geometry. Additional details can be found in [11].

Let $f:M\to {\mathbb{R}}^{n+1}$ be an n-dimensional affine hypersurface. We always assume that f is connected and orientable. Let D be the standard flat connection on ${\mathbb{R}}^{n+1}$. Then, for any transversal vector field C and tangent vector fields $X,Y\in \mathcal{X}\left(M\right)$, we have

$$\begin{array}{c}\hfill D_X{f}_{*}Y=f_{*}\left({\nabla }_{X}Y\right)+h(X,Y)C\end{array}$$

(1)

and

$$\begin{array}{c}\hfill D_{X}C=-f_{*}\left(SX\right)+\tau \left(X\right)C.\end{array}$$

(2)

Here, ∇ is a torsion-free connection; h is a symmetric bilinear form on M, called the second fundamental form; S is a $(1,1)$-tensor, called the shape operator; and $\tau$ is a 1-form.

Throughout this paper, we assume that h is nondegenerate and so defines a pseudo-Riemannian metric on M. We say that the hypersurface is nondegenerate if h is nondegenerate. We have the following:

Theorem 1

([11], Fundamental equations). For an arbitrary transversal vector field C, the induced connection ∇, the second fundamental form h, the shape operator S, and the 1-form τ satisfy the following equations:

$$\begin{array}{cc}& R(X,Y)Z=h(Y,Z)SX-h(X,Z)SY,\hfill \end{array}$$

(3)

$$\begin{array}{cc}& \left({\nabla }_{X}h\right)(Y,Z)+\tau \left(X\right)h(Y,Z)=\left({\nabla }_{Y}h\right)(X,Z)+\tau \left(Y\right)h(X,Z),\hfill \end{array}$$

(4)

$$\begin{array}{cc}& \left({\nabla }_{X}S\right)\left(Y\right)-\tau \left(X\right)SY=\left({\nabla }_{Y}S\right)\left(X\right)-\tau \left(Y\right)SX,\hfill \end{array}$$

(5)

$$\begin{array}{cc}& h(X,SY)-h(SX,Y)=2d\tau (X,Y).\hfill \end{array}$$

(6)

The Equations (3)–(6) are called the equation of Gauss, Codazzi for h, Codazzi for S, and Ricci, respectively.

On an affine hypersurface, one may define a tensor Q of type $(0,3)$ as follows:

$$Q(X_1,X_2,X_3)=\left({\nabla }_{X_1}h\right)(X_2,X_3)+\tau \left(X_1\right)h(X_2,X_3).$$

(7)

The above tensor is called the cubic form. The Formula (4) implies that the cubic form is symmetric with respect to all its variables.

We say that a transversal vector field C is equiaffine if the 1-form $\tau$ vanishes. In the case of $d\tau =0$, we say that C is locally equiaffine.

Let $dimM=2n+1$ and $f:M\to {\mathbb{R}}^{2n+2}$ be a nondegenerate (relative to the second fundamental form) affine hypersurface. We always assume that ${\mathbb{R}}^{2n+2}$ is endowed with the standard paracomplex structure $\tilde{J}$

$$\tilde{J}(x_1,\dots ,x_{n+1},y_1,\dots ,y_{n+1})=(y_1,\dots ,y_{n+1},x_1,\dots ,x_{n+1}).$$

Let C be a transversal vector field on M. We say that C is $\tilde{J}$-tangent if $\tilde{J}{C}_x\in f_{*}\left(T_{x}M\right)$ for every $x\in M$. We also define a distribution $\mathcal{D}$ on M as the biggest $\tilde{J}$-invariant distribution on M, that is

$${\mathcal{D}}_x=f_{*}^{-1}(f_{*}\left(T_{x}M\right)\cap \tilde{J}\left(f_{*}\left(T_{x}M\right)\right))$$

for every $x\in M$. We have that $dim{\mathcal{D}}_x\ge 2n$. If for some x the $dim{\mathcal{D}}_x=2n+1$, then ${\mathcal{D}}_x=T_{x}M$ and it is not possible to find a $\tilde{J}$-tangent transversal vector field in a neighborhood of x. In this paper, we study f with a $\tilde{J}$-tangent transversal vector field C, so in particular $dim\mathcal{D}=2n$. The distribution $\mathcal{D}$ is smooth as the intersection of two smooth distributions and because $dim\mathcal{D}$ is constant. A vector field X is called a $\mathcal{D}$-field if $X_x\in {\mathcal{D}}_x$ for every $x\in M$. We use the notation $X\in \mathcal{D}$ for vectors, as well as for $\mathcal{D}$-fields. We say that the distribution $\mathcal{D}$ is nondegenerate if h is nondegenerate on $\mathcal{D}$.

Additionally, if h is the second fundamental form, one may define a distribution ${\mathcal{D}}_h$ by the following formula

$${{\mathcal{D}}_h}_x:=\{v\in T_{x}M:h(v,w)=0\forall w\in {\mathcal{D}}_x\}.$$

It is clear that ${\mathcal{D}}_h$ is one-dimensional and does not depend on the choice of transversal vector field.

For simplicity, we will usually omit the preceding $f_{*}$, when referring to vector fields.

3. Almost Paracontact Structures

Let M be a differentiable manifold, $dimM=(2n+1)$. A triple $(\phi ,\xi ,\eta )$ is called an almost paracontact structure on M if for every $X\in TM$ the following conditions hold:

$$\begin{array}{cc}\hfill {\phi }^2\left(X\right)& =X-\eta \left(X\right)\xi ,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \eta \left(\xi \right)& =1,\hfill \end{array}$$

(9)

where $\phi$ is a $(1,1)$–tensor, $\xi$ is a vector field, and $\eta$ is a 1-form. Tensor $\phi$ determines on the distribution $\mathcal{D}=ker\eta$ an almost paracomplex structure. The eigenvalues $1,-1$ of $\phi$ correspond to the eigendistributions ${\mathcal{D}}^{+},{\mathcal{D}}^{-}$ and $dim{\mathcal{D}}^{+}=dim{\mathcal{D}}^{-}=n$. If additionally there is a pseudo-Riemannian metric g on M of signature $(n+1,n)$ satisfying for all $X_1,X_2\in TM$

$$\begin{array}{c}\hfill g(\phi X_1,\phi X_2)=-g(X_1,X_2)+\eta \left(X_1\right)\eta \left(X_2\right)\end{array}$$

(10)

then the quadruple $(\phi ,\xi ,\eta ,g)$ is called an almost paracontact metric structure. In particular, for $(\phi ,\xi ,\eta ,g)$ we have

$$\begin{array}{c}\hfill \eta \left(X\right)=g(X,\xi )\end{array}$$

(11)

for all $X\in TM$. Hence, $\xi$ is g-orthogonal to $\mathcal{D}$.

Let us denote by $\widehat{\nabla }$ the Levi-Civita connection for pseudo-metric g. The structure $(\phi ,\xi ,\eta ,g)$ we call a para α-Sasakian structure when for every $X_1,X_2\in \mathcal{X}\left(M\right)$

$$\begin{array}{c}\hfill \left({\widehat{\nabla }}_{X_1}\phi \right)\left(X_2\right)=\alpha (-g(X_1,X_2)\xi +\eta \left(X_2\right)X_1)\end{array}$$

(12)

for some smooth function $\alpha$ on M. In particular, when $\alpha =1$ we obtain the standard para-Sasakian structure. An almost paracontact metric manifold is called para α-contact if

$$\begin{array}{c}\hfill d\eta (X_1,X_2)=\alpha g(X_1,\phi X_2)\end{array}$$

(13)

for a certain non-zero function $\alpha$ and for every $X_1,X_2\in TM$. When $\alpha =1$ is an almost paracontact metric structure $(\phi ,\xi ,\eta ,g)$ satisfying (13) we call this a paracontact metric structure.

An almost paracontact structure $(\phi ,\xi ,\eta )$ satisfying the condition

$$\begin{array}{c}\hfill [\phi ,\phi ]-2d\eta \otimes \xi =0,\end{array}$$

(14)

is called an almost paracontact normal structure. Here, by $[\phi ,\phi ]$ we denote the Nijenhuis tensor for $\phi$. The following result holds

Theorem 2

([12]). An almost paracontact metric manifold is para α-Sasakian if and only if it is normal and para α-contact.

Let C be a transversal vector field for a nondegenerate affine hypersurface $f:M\to {\mathbb{R}}^{2n+2}$. If C is $\tilde{J}$-tangent it is possible to define a $(1,1)$-tensor $\phi$, a vector field $\xi$, and a 1-form $\eta$ by the formulas:

$$\begin{array}{cc}& \xi :=\tilde{J}C;\hfill \end{array}$$

(15)

$$\begin{array}{cc}& {\eta |}_{\mathcal{D}}=0,\eta \left(\xi \right)=1;\hfill \end{array}$$

(16)

$$\begin{array}{cc}& {\phi |}_{\mathcal{D}}=\tilde{J}{|}_{\mathcal{D}},\phi \left(\xi \right)=0.\hfill \end{array}$$

(17)

One can verify that the triple $(\phi ,\xi ,\eta )$ forms an almost paracontact structure on the hypersurface. We will refer to this structure as an induced almost paracontact structure.

The following theorem is now established.

Theorem 3.

Let $(\phi ,\xi ,\eta )$ be an induced almost paracontact structure on an affine hypersurface $f:M\to {\mathbb{R}}^{2n+2}$ with a $\tilde{J}$-tangent transversal vector field C. Then, we have the following identities:

$$\begin{array}{cc}& \phi \left({\nabla }_{X}Y\right)={\nabla }_X\phi Y-\eta \left(Y\right)SX-h(X,Y)\xi ,\hfill \end{array}$$

(18)

$$\begin{array}{cc}& \eta \left({\nabla }_{X}Y\right)=h(X,\phi Y)+X\left(\eta \left(Y\right)\right)+\eta \left(Y\right)\tau \left(X\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}& \eta \left(\right[X,Y\left]\right)=h(X,\phi Y)-h(Y,\phi X)+X\left(\eta \right(Y\left)\right)-Y\left(\eta \right(X\left)\right)\hfill \\ & +\eta \left(Y\right)\tau \left(X\right)-\eta \left(X\right)\tau \left(Y\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}& \phi \left([X,Y]\right)={\nabla }_X\phi Y-{\nabla }_Y\phi X+\eta \left(X\right)SY-\eta \left(Y\right)SX,\hfill \end{array}$$

(21)

$$\begin{array}{cc}& \eta \left({\nabla }_X\xi \right)=\tau \left(X\right),\hfill \end{array}$$

(22)

$$\begin{array}{cc}& \eta \left(SX\right)=-h(X,\xi ),\hfill \end{array}$$

(23)

where $X,Y\in \mathcal{X}\left(M\right)$.

Proof. 

Formulas (15)–(17) imply the following decomposition:

$$\tilde{J}X=\phi X+\eta \left(X\right)C,$$

where $\phi X$ is the tangent part and $\eta \left(X\right)C$ is the transversal part of $\tilde{J}X$, $X\in TM$. Using Formulas (1) and (2), we also have

$$\begin{array}{cc}\hfill \tilde{J}\left(D_{X}Y\right)& =\tilde{J}({\nabla }_{X}Y+h(X,Y)C)=\tilde{J}\left({\nabla }_{X}Y\right)+h(X,Y)\tilde{J}C\hfill \\ \hfill & =\phi \left({\nabla }_{X}Y\right)+\eta \left({\nabla }_{X}Y\right)C+h(X,Y)\xi \hfill \end{array}$$

and

$$\begin{array}{cc}\hfill D_X\tilde{J}Y& =D_X(\phi Y+\eta \left(Y\right)C)=D_X\phi Y+X\left(\eta \left(Y\right)\right)C+\eta \left(Y\right)D_{X}C\hfill \\ \hfill & ={\nabla }_X\phi Y+h(X,\phi Y)C+X\left(\eta \left(Y\right)\right)C+\eta \left(Y\right)(-SX+\tau \left(X\right)C)\hfill \\ \hfill & ={\nabla }_X\phi Y-\eta \left(Y\right)SX+(h(X,\phi Y)+X\left(\eta \left(Y\right)\right)+\eta \left(Y\right)\tau \left(X\right))C.\hfill \end{array}$$

Since $D\tilde{J}=0$, we obtain $D_X\tilde{J}Y=\tilde{J}\left(D_{X}Y\right)$ and in consequence

$$\begin{array}{cc}\hfill \phi \left({\nabla }_{X}Y\right)+\eta \left({\nabla }_{X}Y\right)C+h(X,Y)\xi & ={\nabla }_X\phi Y-\eta \left(Y\right)SX\hfill \\ \hfill & +\left(h\right(X,\phi Y)+X(\eta \left(Y\right))+\eta (Y\left)\tau \right(X\left)\right)C.\hfill \end{array}$$

Now comparing tangent terms

$$\phi \left({\nabla }_{X}Y\right)+h(X,Y)\xi ={\nabla }_X\phi Y-\eta \left(Y\right)SX$$

and transversal terms

$$\eta \left({\nabla }_{X}Y\right)C=(h(X,\phi Y)+X\left(\eta \left(Y\right)\right)+\eta \left(Y\right)\tau \left(X\right))C$$

we obtain formulas (18) and (19), respectively.

From (18) and (19), one immediately obtains the formulas (20)—(23). □

The above theorem allows us to characterize almost paracontact normal structures as follows:

Proposition 1.

Let $(\phi ,\xi ,\eta )$ be an induced almost paracontact structure. The structure $(\phi ,\xi ,\eta )$ is normal if and only if

$$S\phi U-\phi SU+\tau \left(U\right)\xi =0$$

for $U\in \mathcal{D}$

Proof. 

The equivalence is an immediate consequence of (14), the formula

$$d\eta (X_1,X_2)={\displaystyle \frac{1}{2}}(X_1\left(\eta \left(X_2\right)\right)-X_2\left(\eta \left(X_1\right)\right)-\eta \left([X_1,X_2]\right))$$

and (20) and (21). □

Restricting to $\mathcal{D}$-fields, we obtain from Theorem 3 the following corollary:

Corollary 1.

Let $U,V\in \mathcal{D}$, then the following formulas hold:

$$\begin{array}{cc}& \eta \left({\nabla }_{U}V\right)=h(U,\phi V),\hfill \end{array}$$

(24)

$$\begin{array}{cc}& \eta \left({\nabla }_{\xi }U\right)=h(\xi ,\phi U),\hfill \end{array}$$

(25)

$$\begin{array}{cc}& \phi \left({\nabla }_{U}V\right)={\nabla }_U\phi V-h(U,V)\xi ,\hfill \end{array}$$

(26)

$$\begin{array}{cc}& \eta \left(\right[U,V\left]\right)=h(U,\phi V)-h(V,\phi U),\hfill \end{array}$$

(27)

$$\begin{array}{cc}& \eta \left(\right[U,\xi \left]\right)=-h(\xi ,\phi U)+\tau \left(U\right).\hfill \end{array}$$

(28)

4. Main Results

In this section, $(\phi ,\xi ,\eta )$ is considered to be an induced almost paracontact structure and $(\phi ,\xi ,\eta ,h)$ an almost paracontact metric structure. Note that the nondegeneracy of h on $TM$ implies nondegeneracy on $\mathcal{D}$, since $\xi$ is h-orthogonal to $\mathcal{D}$ and $h(\xi ,\xi )=1$ (see (11)).

The following two lemmas are instrumental in proving the main theorem of this section:

Lemma 1.

For an almost paracontact metric structure $(\phi ,\xi ,\eta ,h)$, we have

$$\begin{array}{cc}& \eta \left(X\right)=h(X,\xi ),\mathit{for}\mathit{every}X\in TM,\hfill \end{array}$$

(29)

$$\begin{array}{cc}& S\left(\mathcal{D}\right)\subset \mathcal{D},\hfill \end{array}$$

(30)

$$\begin{array}{cc}& S\xi =-\xi +U_0,\mathit{where}{U}_0\in \mathcal{D}\hfill \end{array}$$

(31)

$$\begin{array}{cc}& \tau \left(U\right)=-h(U,\phi U_0)\mathit{for}\mathit{all}U\in \mathcal{D}.\hfill \end{array}$$

(32)

Proof. 

Properties (29), (30) and (31) are an immediate consequence of (10) and (23). From the Equation (5), we obtain

$${\nabla }_{X}S\xi -S\left({\nabla }_X\xi \right)-\tau \left(X\right)S\xi ={\nabla }_{\xi }SX-S\left({\nabla }_{\xi }X\right)-\tau \left(\xi \right)SX.$$

Formula (25) and the metricity of $(\phi ,\xi ,\eta ,h)$ imply that ${\nabla }_{\xi }U\in \mathcal{D}$ for every $U\in \mathcal{D}$. By (30) and (31) we obtain

$$\begin{array}{c}\hfill \tau \left(U\right)=-\eta \left({\nabla }_U{U}_0\right)+\eta \left({\nabla }_U\xi \right)+\eta \left(S\left({\nabla }_U\xi \right)\right)\end{array}$$

(33)

for all $U\in \mathcal{D}$. By Formulas (19), (23), and (29), and the fact that $(\phi ,\xi ,\eta ,h)$ is metric structure, we obtain

$$\eta \left({\nabla }_U{U}_0\right)=h(U,\phi U_0),\eta \left(S\left({\nabla }_U\xi \right)\right)=-\eta \left({\nabla }_U\xi \right)$$

for all $U\in \mathcal{D}$. As a result, Equation (33) takes the form

$$\tau \left(U\right)=-h(U,\phi U_0).$$

The proof of (32) is concluded. □

Lemma 2.

Let $(\phi ,\xi ,\eta ,h)$ be an almost paracontact metric structure then

$$\begin{array}{cc}& Q(X,U_1,U_2)=-Q(X,\phi U_1,\phi U_2),\hfill \end{array}$$

(34)

$$\begin{array}{cc}& Q(U_1,U_2,U_3)=0,\hfill \end{array}$$

(35)

$$\begin{array}{cc}& Q(\xi ,U,U)=-h(SU,\phi U)=h(S\phi U,U)\hfill \end{array}$$

(36)

for every $X\in \mathcal{X}\left(M\right)$ and $U,U_1,U_2,U_3\in \mathcal{D}$.

Proof. 

First recall that thanks to the Codazzi equation the cubic form Q is symmetric in all three variables. We will make frequent use of this property in proving the lemma. Taking $X\in \mathcal{X}\left(M\right)$ and taking $U_1,U_2\in \mathcal{D}$ by (7) and (10), we have

$$\begin{array}{cc}\hfill Q(X,\phi U_1,\phi U_2)& =X\left(h(\phi U_1,\phi U_2)\right)-h({\nabla }_X\phi U_1,\phi U_2)-h(\phi U_1,{\nabla }_X\phi U_2)\hfill \\ & +\tau \left(X\right)h(\phi U_1,\phi U_2)\hfill \\ \hfill & =-X\left(h(U_1,U_2)\right)-h({\nabla }_X\phi U_1,\phi U_2)-h(\phi U_1,{\nabla }_X\phi U_2)\hfill \\ & -\tau \left(X\right)h(U_1,U_2).\hfill \end{array}$$

From Theorem 3, it follows

$${\nabla }_X\phi U_1=\phi \left({\nabla }_X{U}_1\right)+h(X,U_1)\xi$$

and

$${\nabla }_X\phi U_2=\phi \left({\nabla }_X{U}_2\right)+h(X,U_2)\xi .$$

Thus, using the above and (10), we obtain

$$\begin{array}{cc}\hfill Q(X,\phi U_1,\phi U_2)& =-X\left(h(U_1,U_2)\right)-h(\phi \left({\nabla }_X{U}_1\right),\phi U_2)-h(X,U_1)h(\xi ,\phi U_2)\hfill \\ \hfill & -h(\phi U_1,\phi \left({\nabla }_X{U}_2\right))-h(X,U_2)h(\phi U_1,\xi )-\tau \left(X\right)h(U_1,U_2)\hfill \\ \hfill & =-X\left(h(U_1,U_2)\right)+h({\nabla }_X{U}_1,U_2)+h(U_1,{\nabla }_X{U}_2)-\tau \left(X\right)h(U_1,U_2)\hfill \\ \hfill & =-Q(X,U_1,U_2),\hfill \end{array}$$

thus, (34) is proved. For the proof of (35), note that (34) implies

$$\begin{array}{c}\hfill Q(U,U,U)=-Q(U,\phi U,\phi U)=-Q(\phi U,\phi U,U)=0\end{array}$$

for all $U\in \mathcal{D}$, because $h(\phi U,U)=0$ and ${\nabla }_{\phi U}U=\phi \left({\nabla }_{\phi U}\phi U\right)-h(U,U)\xi$. From the last equation and symmetry of Q in all three variables, we obtain $Q(U_1,U_2,U_3)=0$ for every $U_1,U_2,U_3\in \mathcal{D}$. To prove (36), note first that

$$\begin{array}{c}\hfill Q(\xi ,U,U)=Q(U,\xi ,U)=-h({\nabla }_U\xi ,U)-h(\xi ,{\nabla }_{U}U),\end{array}$$

since $h(\xi ,U)=0$. Formula (18) implies that

$$\phi \left({\nabla }_U\xi \right)=-SU.$$

From (10) and (24), we obtain

$${\nabla }_{U}U\in \mathcal{D}.$$

Now, we have

$$Q(\xi ,U,U)=-h({\nabla }_U\xi ,U)=h(\phi \left(SU\right),U)=-h(SU,\phi U)$$

for all $U\in \mathcal{D}$. The Formula (34) implies that

$$Q(\xi ,U,U)=-Q(\xi ,\phi U,\phi U)$$

and in consequence

$$-h(SU,\phi U)=h(S\phi U,U).$$

Hence, (36) is proved. □

Now the following result will be proved:

Theorem 4.

Let $f:M\to {\mathbb{R}}^{2n+2}$ be a nondegenerate affine hypersurface with a transversal vector field C. If C is $\tilde{J}$-tangent and the induced almost paracontact structure $(\phi ,\xi ,\eta )$ is metric relative to the second fundamental form h, then

$$S=-id\mathit{and}\tau =0.$$

Proof. 

Let $V,U\in \mathcal{D}$. Formulas (5), (16) and (30) imply that

$$\eta \left({\nabla }_{V}SU\right)-\eta \left(S\left({\nabla }_{V}U\right)\right)=\eta \left({\nabla }_{U}SV\right)-\eta \left(S\left({\nabla }_{U}V\right)\right).$$

Thus, by (23) and (11),

$$\eta \left({\nabla }_{V}SU\right)-\eta \left({\nabla }_{U}SV\right)=\eta \left(S\left([V,U]\right)\right)=-\eta \left([V,U]\right)=\eta \left([U,V]\right).$$

By Corollary 1 (the Formulas (24) and (27)), we obtain

$$h(V,\phi SU)-h(U,\phi SV)=-h(V,\phi U)+h(U,\phi V).$$

Substituting U with $\phi U$, we obtain

$$h(V,\phi S\phi U)-h(\phi U,\phi SV)=-h(V,{\phi }^{2}U)+h(\phi U,\phi V).$$

Now making use of the metricity of the structure $(\phi ,\xi ,\eta ,h)$, more precisely the property that

$$h(\phi X,\phi Y)=-h(X,Y),h(\phi X,Y)=-h(X,\phi Y)$$

if at least one of X or Y is in $\mathcal{D}$, and taking into account that ${\phi }^{2}U=U$ for $U\in \mathcal{D}$ and h is symmetric, we obtain

$$-h(\phi V,S\phi U)+h(U,SV)=-2h(V,U)\mathrm{for}\mathrm{all}V,U\in \mathcal{D}.$$

(37)

Formula (3) implies that

$$\begin{array}{cc}\hfill \left(R\right(V,\phi V)·h)(\phi V,\phi V)& =-2h\left(R\right(V,\phi V)\phi V,\phi V)\hfill \\ & =2h(SV,\phi V)h(V,V),\hfill \end{array}$$

(38)

for any $V\in \mathcal{D}$. Alternatively,

$$\begin{array}{c}\hfill (R(V,\phi V)·h)(\phi V,\phi V)=\left({\nabla }_V{\nabla }_{\phi V}h\right)(\phi V,\phi V)\\ \hfill -\left({\nabla }_{\phi V}{\nabla }_{V}h\right)(\phi V,\phi V)-\left({\nabla }_{[V,\phi V]}h\right)(\phi V,\phi V).\end{array}$$

By straightforward computations we obtain

$$\begin{array}{cc}\left({\nabla }_V{\nabla }_{\phi V}h\right)\hfill & (\phi V,\phi V)\hfill \\ \hfill & =V\left(\left({\nabla }_{\phi V}h\right)(\phi V,\phi V)\right)-2\left({\nabla }_{\phi V}h\right)({\nabla }_V\phi V,\phi V),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill \left({\nabla }_{\phi V}{\nabla }_{V}h\right)& (\phi V,\phi V)\hfill \\ \hfill & =\phi V\left(\left({\nabla }_{V}h\right)(\phi V,\phi V)\right)-2\left({\nabla }_{V}h\right)({\nabla }_{\phi V}\phi V,\phi V).\hfill \end{array}$$

(40)

From the definition of the cubic form, we have

$$\begin{array}{c}\hfill \left({\nabla }_{X_1}h\right)(X_2,X_3)=Q(X_1,X_2,X_3)-\tau \left(X_1\right)h(X_2,X_3),\end{array}$$

(41)

for every $X_1,X_2,X_3\in \mathcal{X}\left(M\right)$.

Applying (41) to (39) and using metricity ($h(\phi V,\phi V)=-h(V,V)$), we obtain

$$\begin{array}{cc}\hfill \left({\nabla }_V{\nabla }_{\phi V}h\right)& (\phi V,\phi V)=V\left(Q(\phi V,\phi V,\phi V)-\tau \left(\phi V\right)h(\phi V,\phi V)\right)\hfill \\ & -2Q(\phi V,{\nabla }_V\phi V,\phi V)+2\tau \left(\phi V\right)h({\nabla }_V\phi V,\phi V)\hfill \\ \hfill & =V\left(Q\right(\phi V,\phi V,\phi V\left)\right)-V\left(\tau \right(\phi V\left)\right)h(\phi V,\phi V)\hfill \\ \hfill & -\tau \left(\phi V\right)V\left(h(\phi V,\phi V)\right)-2Q(\phi V,{\nabla }_V\phi V,\phi V)\hfill \\ \hfill & +2\tau \left(\phi V\right)h({\nabla }_V\phi V,\phi V)\hfill \\ \hfill & =V\left(Q\right(\phi V,\phi V,\phi V\left)\right)+V\left(\tau \right(\phi V\left)\right)h(V,V)\hfill \\ \hfill & -\tau \left(\phi V\right)V\left(h(\phi V,\phi V)\right)-2Q(\phi V,{\nabla }_V\phi V,\phi V)\hfill \\ \hfill & +2\tau \left(\phi V\right)h({\nabla }_V\phi V,\phi V)\hfill \end{array}$$

First note that Formula (35) from Lemma 2 implies that

$$Q(\phi V,\phi V,\phi V)=0.$$

Using symmetry of Q and Formula (34) from Lemma 2, we also obtain

$$\begin{array}{cc}\hfill Q(\phi V,{\nabla }_V\phi V,\phi V)& =Q({\nabla }_V\phi V,\phi V,\phi V)=-Q({\nabla }_V\phi V,V,V)\hfill \\ \hfill & =-Q(\xi ,V,V)h(V,V),\hfill \end{array}$$

where the last equality follows from the fact that ${\nabla }_V\phi V$ can be decomposed as follows:

$${\nabla }_V\phi V=D_0+\eta \left({\nabla }_V\phi V\right)\xi =D_0+h(V,V)\xi ,$$

since $\eta \left({\nabla }_V\phi V\right)=h(V,V)$ thanks to (24). Now, we obtain

$$\begin{array}{cc}\hfill \left({\nabla }_V{\nabla }_{\phi V}h\right)(\phi V,\phi V)& =V\left(\tau \right(\phi V\left)\right)h(V,V)-\tau \left(\phi V\right)V\left(h\right(\phi V,\phi V\left)\right)\hfill \\ \hfill & +2h(V,V)Q(\xi ,V,V)+2\tau \left(\phi V\right)h({\nabla }_V\phi V,\phi V)\hfill \\ \hfill & =V\left(\tau \right(\phi V\left)\right)h(V,V)\hfill \\ \hfill & -\tau \left(\phi V\right)\left(V\left(h(\phi V,\phi V)\right)-2h({\nabla }_V\phi V,\phi V)\right)\hfill \\ \hfill & +2h(V,V)Q(\xi ,V,V)\hfill \\ \hfill & =V\left(\tau \left(\phi V\right)\right)h(V,V)-\tau \left(\phi V\right)\left({\nabla }_{V}h\right)(\phi V,\phi V)\hfill \\ \hfill & +2h(V,V)Q(\xi ,V,V).\hfill \end{array}$$

Since

$$\begin{array}{cc}\hfill \left({\nabla }_{V}h\right)(\phi V,\phi V)& =Q(V,\phi V,\phi V)-\tau \left(V\right)h(\phi V,\phi V)\hfill \\ \hfill & =\tau \left(V\right)h(V,V)\hfill \end{array}$$

we conclude that

$$\begin{array}{cc}\hfill \left({\nabla }_V{\nabla }_{\phi V}h\right)(\phi V,\phi V)& =V\left(\tau \right(\phi V\left)\right)h(V,V)-\tau \left(\phi V\right)\tau \left(V\right)h(V,V)\hfill \\ & +2h(V,V)Q(\xi ,V,V).\hfill \end{array}$$

Similarly, applying (41) to (40) and using Lemma 2 we find that

$$\begin{array}{cc}\hfill \left({\nabla }_{\phi V}{\nabla }_{V}h\right)(\phi V,\phi V)& =\phi V\left(\tau \right(V\left)\right)h(V,V)-\tau \left(V\right)\tau \left(\phi V\right)h(V,V).\hfill \end{array}$$

By (27) and Lemma 2 we also have

$$\begin{array}{cc}\hfill \left({\nabla }_{[V,\phi V]}h\right)(\phi V,\phi V)& =Q\left(\right[V,\phi V],\phi V,\phi V)-\tau \left(\right[V,\phi V\left]\right)h(\phi V,\phi V)\hfill \\ \hfill & =-\eta \left(\right[V,\phi V\left]\right)Q(\xi ,V,V)+\tau \left(\right[V,\phi V\left]\right)h(V,V)\hfill \\ \hfill & =-2h(V,V)Q(\xi ,V,V)+\tau \left(\right[V,\phi V\left]\right)h(V,V).\hfill \end{array}$$

Combining (36) with (6), we derive

$$\begin{array}{c}\hfill 2Q(\xi ,V,V)=-h(SV,\phi V)+h(V,S\phi V)=2d\tau (V,\phi V).\end{array}$$

(42)

Now, it follows from (42) and the previous expressions that

$$\left(R\right(V,\phi V)·h)(\phi V,\phi V)=6d\tau (V,\phi V)h(V,V)$$

thus, using (42) and (36),

$$\left(R\right(V,\phi V)·h)(\phi V,\phi V)=6Q(\xi ,V,V)h(V,V)=-6h(V,V)h(SV,\phi V),$$

which, together with (38), gives for all $V\in \mathcal{D}$

$$h(V,V)·h(SV,\phi V)=0.$$

Using the fact that h is nondegenerate on $\mathcal{D}$, we obtain

$$h(SV,\phi V)=0$$

(43)

for all $V\in \mathcal{D}$. Eventually, from Formula (43), it follows that

$$0=h\left(S\right(V+2\phi U),\phi V+2U)=2h(SV,U)+2h(S\phi U,\phi V).$$

Therefore,

$$h(S\phi U,\phi V)=-h(SV,U).$$

By (37) we also have

$$h(S\phi U,\phi V)=2h(V,U)+h(SV,U).$$

The above formulas imply that

$$h(SV,U)=-h(V,U)$$

for all $U\in \mathcal{D}$. Now, since $\mathcal{D}$ is nondegenerate, the Formula (30) implies

$$SV=-V$$

for all $V\in \mathcal{D}$. Therefore, for any $X\in X\left(M\right)$, by Lemma 1, we obtain

$$\begin{array}{c}\hfill SX=-X+\eta \left(X\right)U_0.\end{array}$$

(44)

Now, we intend to prove that $U_0=0$. For this purpose, suppose that $U_0\ne 0$. From the Equation (5), it follows that

$${\nabla }_{V}S{U}_0-S\left({\nabla }_V{U}_0\right)-\tau \left(V\right)S{U}_0={\nabla }_{U_0}SV-S\left({\nabla }_{U_0}V\right)-\tau \left(U_0\right)SV.$$

Since $\tau \left(U_0\right)=0$ (Lemma 1), by means of (44), the above equality can be rewritten as follows:

$$-\eta \left({\nabla }_V{U}_0\right)U_0+\tau \left(V\right)U_0=-\eta \left({\nabla }_{U_0}V\right)U_0,$$

Now, by (27) and (32), we have

$$-\tau \left(V\right)U_0=\eta \left([U_0,V]\right)U_0=-2h(V,\phi U_0)U_0=2\tau \left(V\right)U_0.$$

We deduce from the above equation that $\tau =0$ on $\mathcal{D}$. Furthermore, Equation (32) yields $U_0=0$, contradicting the initial postulate.

From the identity $S=-id$ and the Codazzi equation, it follows straightforwardly that $\tau =0$. The proof is completed. □

The following theorem gives equivalent conditions for being the induced almost paracontact metric structure.

Theorem 5.

Let $f:M\to {\mathbb{R}}^{2n+2}$ be a nondegenerate affine hypersurface with a $\tilde{J}$-tangent transversal vector field and let $(\phi ,\xi ,\eta )$ be the induced almost paracontact structure on M. The following statements are equivalent:

$$\begin{array}{cc}\hfill & (\phi ,\xi ,\eta ,h)\mathit{is}\mathit{an}\mathit{almost}\mathit{paracontact}\mathit{metric}\mathit{structure},\hfill \end{array}$$

(45)

$$\begin{array}{cc}& (\phi ,\xi ,\eta ,h)\mathit{is}a\mathit{para}\alpha \text{-}\mathit{contact}\mathit{metric}\mathit{structure},\mathit{where}\alpha =-1,\hfill \end{array}$$

(46)

$$\begin{array}{cc}& (\phi ,\xi ,\eta ,h)\mathit{is}\mathit{a}\mathit{para}\alpha \text{-}\mathit{Sasakian}\mathit{structure},\mathit{where}\alpha =-1.\hfill \end{array}$$

(47)

Proof. 

In order to prove the implication (45) ⇒ (46), first note that the metricity of $(\phi ,\xi ,\eta ,h)$ and Theorem 4 imply that $\tau =0$. Now, using (20) and (10), we obtain

$$2d\eta (X_1,X_2)=h(X_2,\phi X_1)-h(X_1,\phi X_2)=-2h(X_1,\phi X_2),$$

where $X_1,X_2\in TM$. In consequence, $(\phi ,\xi ,\eta ,h)$ is para $(-1)$-contact. The implication (46) ⇒ (47) follows from Theorem 2 and Proposition 1, since $S=-id$ and $\tau =0$ (thus the structure is normal). The implication (47) ⇒ (45) is obvious. □

Using the Pick–Berwald theorem, we obtain

Theorem 6.

Let $f:M\to {\mathbb{R}}^{2n+2}$ be a nondegenerate affine hypersurface equipped with a $\tilde{J}$-tangent transversal vector field. Let $(\phi ,\xi ,\eta )$ be the induced almost paracontact structure on M. If the structure $(\phi ,\xi ,\eta ,h)$ is metric, then the image $f\left(M\right)$ lies on a hyperquadric.

Proof. 

First, note that Theorem 4 implies $S=-id$ and $\tau =0$. From Lemma 2 (Formula (35)), it follows that

$$\begin{array}{c}\hfill Q(W_1,W_2,W_3)=0\end{array}$$

(48)

for all $W_1,W_2,W_3\in \mathcal{D}$. Since Q is symmetric, $S=-id$ and $h(\phi W,W)=0$ for any $W\in \mathcal{D}$ the Lemma 2 (Formula (36)) implies that

$$\begin{array}{c}\hfill Q(\xi ,W_1,W_2)=0\end{array}$$

(49)

for all $W_1,W_2\in \mathcal{D}$. Due to the fact that $\tau =0$, using (29) and (19) we also obtain

$$\begin{array}{c}\hfill Q(X,\xi ,\xi )=-2h({\nabla }_X\xi ,\xi )=-2\eta \left({\nabla }_X\xi \right)=0\end{array}$$

(50)

for every $X\in \mathcal{X}\left(M\right)$.

Now, let $X,Y,Z$ be arbitrary vector fields on M. Each of them can be decomposed as follows:

$$X=X_D+{\alpha }_X\xi ,Y=Y_D+{\alpha }_Y\xi ,Z=Z_D+{\alpha }_Z\xi ,$$

where $X_D,Y_D,Z_D\in \mathcal{D}$ and ${\alpha }_X,{\alpha }_Y,{\alpha }_Z$ are some smooth functions. Since Q is symmetric in all three variables, we obtain

$$\begin{array}{cc}\hfill Q(X,Y,Z)& =Q(X_D,Y_D,Z_D)+{\alpha }_{Z}Q(X_D,Y_D,\xi )+{\alpha }_{Y}Q(X_D,\xi ,Z_D)\hfill \\ \hfill & +{\alpha }_{X}Q(\xi ,Y_D,Z_D)+{\alpha }_Y{\alpha }_{Z}Q(X_D,\xi ,\xi )+{\alpha }_X{\alpha }_{Z}Q(\xi ,Y_D,\xi )\hfill \\ \hfill & +{\alpha }_X{\alpha }_{Y}Q(\xi ,\xi ,Z_D)+{\alpha }_X{\alpha }_Y{\alpha }_{Z}Q(\xi ,\xi ,\xi ).\hfill \end{array}$$

Now, from (48)–(50) it follows that all the above components are equal to zero. Thus, $Q\equiv 0$ and the Pick–Berwald theorem imply that the hypersurface $f\left(M\right)$ is a piece of a hyperquadric. □

Finally, we can find an explicit formula for such hyperquadrics. We have the following theorem:

Theorem 7.

The nondegenerate hyperquadric of center 0 such that the induced almost paracontact structure $(\phi ,\xi ,\eta )$ is metric relative to the second fundamental form h can be expressed in the form

$$\begin{array}{c}\hfill H=\{x\in {\mathbb{R}}^{2n+2}:x^{T}Ax=1\},\end{array}$$

where $detA\ne 0$ and

$$A=\left[\begin{array}{cc}P& R\\ -R& -P\end{array}\right],$$

$P^T=P$, $R^T=-R$, $P,R\in M(n+1,n+1,\mathbb{R})$.

Moreover, the induced almost paracontact structure for hyperquadrics of the above form is metric with respect to h.

Proof. 

Every nondegenerate hyperquadric of center 0 has a form

$$\begin{array}{c}\hfill H=\{x\in {\mathbb{R}}^{2n+2}:x^{T}Ax=1\},\end{array}$$

where $detA\ne 0$, $A^T=A$, $A\in M(2n+2,2n+2,\mathbb{R})$. From Theorem 4 (the structure $(\phi ,\xi ,\eta )$ is metric) we obtain $S=-id$, $\tau =0$. So $C:=x$ is $\tilde{J}$-tangent and $\xi =\tilde{J}x$ is tangent. Since $Ax$ is orthogonal (relative to the standard inner product $⟨·,·⟩$ on ${\mathbb{R}}^{2n+2}$) to H, we have

$$0=⟨\tilde{J}x,Ax⟩=x^T\tilde{J}Ax$$

for every $x\in H$. We also have

$$x^{T}A\tilde{J}x=0,$$

so in consequence

$$\begin{array}{cc}& x^T(\tilde{J}A+A\tilde{J})x=0\hfill \end{array}$$

(51)

for every $x\in H$. Since $\tilde{J}A+A\tilde{J}$ is symmetric, Formula (51) implies that

$$\begin{array}{c}\hfill \tilde{J}A=-A\tilde{J}.\end{array}$$

The last formula implies that

$$A=\left[\begin{array}{cc}P& R\\ -R& -P\end{array}\right],$$

$P^T=P$, $R^T=-R$, $P,R\in M(n+1,n+1,\mathbb{R})$.

In order to prove the second part of the theorem, note that since $⟨Ax,C⟩=1$, it is sufficient to show that

$$\begin{array}{c}\hfill ⟨D_{\tilde{J}Z}\tilde{J}W,Ax⟩=-⟨D_{Z}W,Ax⟩\end{array}$$

for every $Z,W\in \mathcal{D}$. For $Z,W\in \mathcal{D}$ we have

$$\begin{array}{cc}& ⟨Z,Ax⟩=0,\hfill \end{array}$$

(52)

$$\begin{array}{cc}& ⟨\tilde{J}Z,Ax⟩=0,\hfill \end{array}$$

(53)

$$\begin{array}{cc}& ⟨W,Ax⟩=0,\hfill \end{array}$$

(54)

$$\begin{array}{cc}& ⟨\tilde{J}W,Ax⟩=0.\hfill \end{array}$$

(55)

We also have

$$\begin{array}{cc}& ⟨\tilde{J}X,Y⟩=⟨X,\tilde{J}Y⟩\hfill \end{array}$$

(56)

for every $X,Y$ tangent to hyperquadric. Using the fact that $DA=0$, we obtain

$$\begin{array}{cc}& D_{Z}Ax=AZ\hfill \end{array}$$

(57)

and

$$\begin{array}{cc}& D_{\tilde{J}Z}Ax=A\tilde{J}Z.\hfill \end{array}$$

(58)

Since $D⟨·,·⟩=0$, we also have

$$⟨D_{\tilde{J}Z}\tilde{J}W,Ax⟩=\tilde{J}W\left(⟨\tilde{J}Z,Ax⟩\right)-⟨\tilde{J}W,D_{\tilde{J}Z}Ax⟩.$$

Using (53), (58) and (56), we obtain

$$\begin{array}{c}\hfill ⟨D_{\tilde{J}Z}\tilde{J}W,Ax⟩=-⟨\tilde{J}W,A\tilde{J}Z⟩=⟨\tilde{J}W,\tilde{J}AZ⟩=⟨W,AZ⟩\end{array}$$

On the other hand, (57), (54) and $D⟨·,·⟩=0$ imply

$$⟨W,AZ⟩=⟨W,D_{Z}Ax⟩=-⟨D_{Z}W,Ax⟩,$$

which completes the proof. □

We conclude this section with the following explicit example

Example 1.

Let $\tilde{J}$ be the standard paracomplex structure on ${\mathbb{R}}^4$:

$$\tilde{J}(x_1,x_2,x_3,x_4)=(x_3,x_4,x_1,x_2).$$

Let us consider the 3-dimensional hyperquadric given by the equation

$$\begin{array}{c}\hfill F(x_1,x_2,x_3,x_4)=1,\end{array}$$

(59)

where

$$F(x_1,x_2,x_3,x_4)=x_1^2+x_2^2-x_3^2-x_4^2+2x_1{x}_4-2x_2{x}_3.$$

Since

$$gradF=\left[\begin{array}{c}2(x_1+x_4)\\ 2(x_2-x_3)\\ 2(-x_2-x_3)\\ 2(x_1-x_4)\end{array}\right]$$

we easily compute that $⟨gradF,\tilde{J}C⟩=0$, where $C=(x_1,x_2,x_3,x_4)$. Thus, C is a $\tilde{J}$-tangent transversal vector field. By straightforward computations, one may check that the vector fields

$$D_1=\left[\begin{array}{c}{x}_1+x_2-x_3-x_4\\ -x_1+x_2+x_3-x_4\\ x_1+x_2-x_3-x_4\\ -x_1+x_2+x_3-x_4\end{array}\right],D_2=\left[\begin{array}{c}{x}_1-x_2+x_3-x_4\\ x_1+x_2+x_3+x_4\\ -(x_1-x_2+x_3-x_4)\\ -(x_1+x_2+x_3+x_4)\end{array}\right],\xi =\left[\begin{array}{c}{x}_3\\ x_4\\ x_1\\ x_2\end{array}\right]$$

form a basis on the hyperquadric. Moreover, since $\tilde{J}{D}_1=D_1$ and $\tilde{J}{D}_2=-D_2$, the vector fields $D_1,D_2$ span the distribution $\mathcal{D}$. One may also show that the second fundamental form h in the above basis takes the form

$$h=\left[\begin{array}{ccc}0& 4& 0\\ 4& 0& 0\\ 0& 0& 1\end{array}\right]$$

Thus, the induced structure $(\phi ,\xi ,\eta ,h)$ is an almost paracontact metric structure.

Note that the hyperquadric (59) can be rewritten in the form

$$x^{T}Ax=1$$

where

$$A=\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& -1& 0\\ 0& -1& -1& 0\\ 1& 0& 0& -1\end{array}\right]$$

which agrees with Theorem 7.

5. Conclusions

This paper provided a connection between paracontact/paracomplex geometry and affine differential geometry. In the paper, we investigated the conditions under which a naturally induced almost paracontact structure on an affine hypersurface is metric with respect to the second fundamental form. It appears that there are strong constraints on affine hypersufraces admitting a metric almost paracontact structure. As shown in the central theorem of the article (Theorem 4), metricity implies that a $\tilde{J}$-tangent transversal vector field must be equiaffine and the shape operator has minus identity.

The paper also states an equivalence relation between almost paracontact metric structures, para $\alpha$-contact metric structures, and para $\alpha$-Sasakian structures ($\alpha =-1$). Furthermore, we show that the cubic form must vanish on such hypersurfaces and, in consequence, thanks to the Pick–Berwald theorem, these hypersufaces are pieces of hyperquadrics. Finally, we provide a complete classification of such hypersufaces and find explicit formulas for them (Theorem 7).