Abstract
In this paper we study affine hypersurfaces with non-degenerate second fundamental form of arbitrary signature additionally equipped with an almost symplectic structure \(\omega \). We prove that if \(R^p\omega =0\) or \(\nabla ^p\omega =0\) for some positive integer p then the rank of the shape operator is at most one. The results provide complete classification of affine hypersurfaces with higher order parallel almost symplectic forms and are generalization of recently obtained results for Lorentzian affine hypersurfaces.
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1 Introduction
Parallel structures are of the great interest in the classical Riemannian geometry (see [3, 6, 17]) as well as in affine differential geometry [4, 8, 9, 12,13,14,15]. Higher order parallel structures are the natural generalization of parallel structures and are widely studied as well [6, 7, 23,24,25].
In [2] O. Baues and V. Cortés studied affine hypersurfaces equipped with an almost complex structure.
They showed that there is direct relation between simply connected special Kähler manifolds [10] and improper affine hyperspheres. Later V. Cortés together with M.-A. Lawn and L. Schäfer proved a similar result for special para-Kähler manifolds [5]. In both cases an important role was played by the Kählerian (resp. para-Kählerian) symplectic form \(\omega \). The concept of special affine hyperspheres was generalized by the first author in [21]. Some other results related to affine hypersurfaces with almost complex structures can be also found in the paper of M. Kon [16]. In all the above cases the important role was played by an (almost) symplectic structure related in some way to the induced affine structure on a hypersurface. In particular, relation between an almost symplectic structure \(\omega \) and the induced affine connection \(\nabla \) and its curvature R seemed crucial.
The above results motivated the first author to study non-degenerate affine hypersurfaces \(f:M\rightarrow \mathbb {R}^{2n+1}\) with a transversal vector field \(\xi \) additionally equipped with an almost symplectic structure \(\omega \) in a more general setting. More precisely affine hypersurfaces with the property \(\nabla ^p\omega =0\) or even more general \(R^p\omega =0\). In [19] it was shown that if \(\dim M\ge 4\) condition \(R\omega =0\) implies that \(\nabla \) must be flat (what is generalization of result obtained in [16]) and the result generalizes to an arbitrary power of R under additional assumption that the second fundamental form is positive definite and the transversal vector field \(\xi \) is locally equiaffine (i.e. \(d\tau =0\)). Namely, we have
Theorem 1.1
([19]). Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) be a non-degenerate affine hypersurface (\(\dim M\ge 4\)) with a locally equiaffine transversal vector field \(\xi \) and an almost symplectic form \(\omega \). Additionally assume that the second fundamental form is positive definite on M. If \(R^p\omega =0\) for some positive integer p then \(\nabla \) is flat.
From the above theorem it follows that
Theorem 1.2
([19]). Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) be a non-degenerate affine hypersurface (\(\dim M\ge 4\)) with a locally equiaffine transversal vector field \(\xi \) and an almost symplectic form \(\omega \). Additionally assume that the second fundamental form is positive definite on M. If \(\nabla ^p\omega =0\) for some positive integer p then \(\nabla \) is flat.
In the very same paper it was shown that the condition that the second fundamental form is positive definite cannot be relaxed since there exists affine hypersurface with property \(R^2\omega =0\) which is not flat.
Later in [20] it was shown that for affine hypersurfaces with Lorentzian second fundamental form we still have strong constrains on the shape operator S if only \(\dim M\ge 6\). More precisely, although it cannot be shown that \(S=0\) (what is equivalent with \(\nabla \) being flat) one may still show that \({\text {rank}}S\le 1\). Recently ([22]) it was shown that the same constrains apply when \(\dim M = 4\). Combining results from [20, 22] we have the following theorems:
Theorem 1.3
([20, 22]). Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) (\(\dim M\ge 4\)) be a non-degenerate affine hypersurface with a locally equiaffine transversal vector field \(\xi \) and an almost symplectic form \(\omega \). If \(R^p\omega =0\) for some \(p\ge 1\) and the second fundamental form is Lorentzian on M (that is has signature \((2n-1,1)\)) then the shape operator S has the rank \(\le 1\).
Theorem 1.4
([20, 22]). Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) (\(\dim M\ge 4\)) be a non-degenerate affine hypersurface with a locally equiaffine transversal vector field \(\xi \) and an almost symplectic form \(\omega \). If \(\nabla ^p\omega =0\) for some \(p\ge 1\) and the second fundamental form is Lorentzian on M (that is has signature \((2n-1,1)\)) then the shape operator S has the rank \(\le 1\).
The main purpose of the present paper is to prove that the condition that the second fundamental form is Lorentzian can be dropped and Theorems 1.3 and 1.4 stay true for an arbitrary non-degenerate second fundamental form. The main difficulty of this generalization comes from the fact that due to exponential grow of different possible scenarios methods from [20,21,22] cannot be easily repeated. For this reasons we need to change approach and first focus more on particular types of Jordan blocks rather than on all possible configurations. Reducing number of allowed Jordan blocks dramatically decrease number of different configurations we need to consider when proving main theorems of this paper.
The paper is organized as follows. In Sect. 2 we briefly recall the basic formulas of affine differential geometry, Jordan decomposition and some basic definitions from symplectic geometry. In this section we also prove a simple but important lemma about simultaneous decomposition of the shape operator S and the second fundamental form h. The Sect. 3 is devoted to real Jordan blocks. The main result of this section is that the Jordan decomposition of the shape operator cannot contain real Jordan blocks of dimension grater than or equal 3 and it contains at most one block of dimension 2. In Sect. 4 we study complex Jordan blocks. It is shown that condition \(R^p\omega =0\) implies that the Jordan decomposition of the shape operator cannot contain complex Jordan blocks. Section 5 contains the main results of this paper. Basing on results from Sects. 3 and 4, we show that if there exists an almost symplectic structure \(\omega \) satisfying condition \(R^p\omega =0\) or \(\nabla ^p\omega =0\) for some positive integer p then the rank of the shape operator must be \(\le 1\). We conclude the section with some general example.
2 Preliminaries
We briefly recall the basic formulas of affine differential geometry. For more details, we refer to [18]. Let \(f:M\rightarrow \mathbb {R}^{n+1}\) be an orientable connected differentiable n-dimensional hypersurface immersed in the affine space \(\mathbb {R}^{n+1}\) equipped with its usual flat connection \({\text {D}}\). Then for any transversal vector field \(\xi \) we have
and
where X, Y are vector fields tangent to M. It is known that \(\nabla \) is a torsion-free connection, h is a symmetric bilinear form on M, called the second fundamental form, S is a tensor of type (1, 1), called the shape operator, and \(\tau \) is a 1-form, called the transversal connection form. The vector field \(\xi \) is called equiaffine if \(\tau =0\). When \(d\tau =0\) the vector field \(\xi \) is called locally equiaffine.
When h is non-degenerate then h defines a pseudo-Riemannian metric on M. In this case we say that the hypersurface or the hypersurface immersion is non-degenerate. In this paper we always assume that f is non-degenerate. We have the following
Theorem 2.1
([18], Fundamental equations). For an arbitrary transversal vector field \(\xi \) the induced connection \(\nabla \), the second fundamental form h, the shape operator S, and the 1-form \(\tau \) satisfy the following equations:
The Eqs. (3), (4), (5), and (6) are called the equations of Gauss, Codazzi for h, Codazzi for S and Ricci, respectively.
Let \(\omega \) be a non-degenerate 2-form on manifold M. The form \(\omega \) we call an almost symplectic structure. It is easy to see that if a manifold M admits some almost symplectic structure then M is orientable manifold of even dimension. Structure \(\omega \) is called a symplectic structure, if it is almost symplectic and additionally satisfies \(d\omega =0\). Pair \((M,\omega )\) we call a (an almost) symplectic manifold, if \(\omega \) is a (an almost) symplectic structure on M.
Recall ([1]) that an affine connection \(\nabla \) on an almost symplectic manifold \((M,\omega )\) we call an almost symplectic connection if \(\nabla \omega =0\). An affine connection \(\nabla \) on an almost symplectic manifold \((M,\omega )\) we call a symplectic connection if it is almost symplectic and torsion-free.
Now we recall a well-know theorem about Jordan normal form (See eg. Th. A.2.6 in [11]).
Theorem 2.2
([11], Jordan). If \(A:V\rightarrow V\) is an endomorphism of real finite dimensional vector space V then there exists a basis of V such that the matrix of the endomorphism A in this basis has a form
where \(L_i\) is the Jordan block corresponding to the eigenvalue \(\lambda _i\) and given by the formula
when \(\lambda _i\) is a real number, or by the formula
where
when \(\lambda _i=\alpha _i+i\beta _i\) (\(\beta _i\ne 0\)) is a complex number.
A square matrix P of dimension n is called the sip matrix (the standard involutory permutation) [11] if it has a form:
Note that P is non-singular symmetric matrix and \(P^2=I\). In particular all its eigenvalues are equal \(\pm 1\). Moreover, it is easy to verify that we have the following formula for signature of P:
Theorem 2.3
(Th. 6.1.5 [11]). Let H be a real invertible and symmetric matrix of dimension n. Then for every square n dimensional and H-selfadjoint matrix A (i.e. \(A^{T}H=HA\)) there exists a basis \(\{e_1,\ldots ,e_n\}\) such that
where \(J_1,\ldots ,J_t\) are Jordan blocks of type (8) and \(J_{t+1},\ldots ,J_{t+s}\) are Jordan blocks of type (9). Moreover
where \(P_j\) is a sip matrix of dimension equal to dimension of matrix \(J_j\) for \(j=1,\ldots ,t+s\) and \(\varepsilon _j=\pm 1\) for \(j=1,\ldots ,t\). The signs \(\varepsilon _j\) are determined uniquely by (A, H) up to permutation of signs in the blocks of (13) corresponding to the Jordan blocks of A with the same real eigenvalue and the same size.
For a tensor field T of type (0, p) its covariant derivation \(\nabla T\) is a tensor field of type \((0,p+1)\) given by the formula:
Higher order covariant derivatives of T can be defined by recursion:
To simplify computation it is often convenient to define \(\nabla ^0T:= T\).
If R is a curvature tensor for an affine connection \(\nabla \), one can define a new tensor \(R\cdot T\) of type \((0,p+2)\) by the formula
Analogously to the previous case, we may define a tensor \(R^k\cdot T\) of type \((0,2k+p)\) using the following recursive formula:
and additionally \(R^0\cdot T:=T\).
In order to simplify the notation, we will be often omitting “\(\cdot \)” in \(R^k\cdot T\) when no confusion arises. Thus we will be writing often \(R^k T\) instead of \(R^k\cdot T\).
We conclude this section with the following lemma:
Lemma 2.4
Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) be a non-degenerate affine hypersurface with a locally equiaffine transversal vector field \(\xi \). Then for every point \(x\in M\) there exists a basis \({e_1,\ldots ,e_{2n}}\) of \(T_xM\) such that the shape operator S and the second fundamental form h can be expressed in this basis in the block matrix form
and \(S_i\), \(H_i\) satisfy the following conditions:
-
For \(i=1,\ldots ,q+r\) \(\dim S_i=\dim H_i\).
-
For \(i=1,\ldots ,q\) \(S_i\) is a Jordan block of type (8) and \(\dim S_i\ge \dim S_{i+1}\) for \(i=1,\ldots ,q-1\).
-
For \(i=q+1,\ldots ,q+r\) \(S_i\) is a Jordan block of type (9) and \(\dim S_i\ge \dim S_{i+1}\) for \(i=q+1,\ldots ,q+r-1\).
-
For \(i=1,\ldots ,q\) \(H_i\) is up to a sign sip matrix.
-
For \(i=q+1,\ldots ,q+r\) \(H_i\) is a sip matrix.
Proof
Since \(\xi \) is locally equiaffine we have \(d\tau =0\) and in consequence \(h(SX,Y)=h(X,SY)\) for all \(X,Y\in T_xM\). Now thesis immediately follows from Theorem 2.3 and the fact that we can rearrange Jordan blocks \(S_i\) and matrices \(H_i\) in desired order (rearranging vectors \({e_1,\ldots ,e_{2n}}\) if needed). \(\square \)
3 Real Jordan Blocks
In this chapter we study properties of real Jordan blocks of the shape operator S.
In all the below lemmas we assume that \(f:M\rightarrow \mathbb {R}^{2n+1}\) is a non-degenerate affine hypersurface with a locally equiaffine transversal vector field \(\xi \) and an almost symplectic form \(\omega \). About objects \(\nabla \), h, S and \(\tau \) we assume that they are induced by \(\xi \).
In the following lemmas (if not stated otherwise) we assume that \(S_1\) (from Lemma 2.4) is a k-dimensional block of the form
where \(\alpha \in \mathbb {R}\) and
where \(\varepsilon \in \{-1,1\}\). By \(\{e_1,\ldots ,e_{2n}\}\) we will be denoting basis of \(T_xM\) such that \(\{e_1,\ldots ,e_{k}\}\) is a basis for \(S_1\).
Lemma 3.1
Let \(p\ge 1\). If \(k\ge 2\) and \(X_1,\ldots ,X_p\in {\text {span}} \{e_1,\ldots ,e_k\}=:V\) then for every \(i\in \{2,3,\ldots ,2n\}\)
where \(\pi :V\rightarrow \mathbb {R}\) is a projection defined as follows:
if \(X=\lambda _1e_1+\ldots +\lambda _ke_k\in V\) for some \(\lambda _1,\ldots ,\lambda _k\in \mathbb {R}\).
Proof
First we shall prove the following formulas:
for all \(X,Y \in V\). To prove (17) we compute
To prove (18) we compute
To prove (19) we compute
Now using formulas (17), (18), (19) we shall prove the thesis of the lemma. For \(p=1\) we have
Assume that formula (16) is true for some \(p\ge 1\). Then for \(p+1\) we get
Now, by the induction principle the formula (16) holds for every \(p\ge 1\). \(\square \)
As an immediate consequence of Lemma 3.1 (setting \(X_1=X_2=\ldots =X_p=e_1\)) we obtain:
Corollary 3.2
If \(k\ge 2\) then for every \(i\in \{2,\ldots ,2n\}\) we have
In the next few lemmas we shall obtain some properties of \(R^p\omega \) under the assumption that \(k>3\).
Lemma 3.3
If \(k>3\) then
Proof
The proof is an immediate consequence of the Gauss equation and the fact that
and
if only \(k>3\). \(\square \)
Lemma 3.4
If \(k>3\) and \(p\ge 1\) we have
Proof
First note that the formula (26) easily follows from (20) for every \(p\ge 1\).
In order to prove (27) let us note that for \(p=1\) we have
thanks to (20) and (21). Now assume that (27) is true for some \(p\ge 1\). Using (20), (21) and (26) we compute
Thus, by the induction principle (27) is true for all \(p\ge 1\).
In order to prove (28) first note that if \(e_i\bot \{e_1,e_{k-1}\}\) we have
In particular the above holds for all \(i\in \{1,\ldots ,2n\}\setminus \{2,k\}\). Using (22) and (29) we get that for every \(i\in \{1,\ldots ,2n\}\setminus \{2,k\}\)
thus (28) is true for \(p=1\). Now assume that (28) holds for some \(p\ge 1\) and all \(i\in \{1,\ldots ,2n\}\setminus \{2,k\}\). We have
since
by (20) and (29). Now, by the induction principle (28) holds for all \(p\ge 1\). \(\square \)
Lemma 3.5
If \(k>3\) and \(p\ge 0\) we have
Proof
First note that by straightforward computations, using (20), (21) and (22), one may easily check that (30) and (31) are true for \(p=0\).
Let us assume that \(p>0\). In order to prove (30), using Lemma 3.3, we compute
Now using (28) (for \(i=1\)) and (27) we obtain
that is (30) is true for all \(p\ge 0\).
In order to prove (31) we compute
where the last equality follows from (20). Now using (21) and (22) we get
Again using (28) (for \(i=1\)) and (27) we obtain
what completes the proof of (31). \(\square \)
Lemma 3.6
If \(k>3\) and \(p\ge 1\) we have
Proof
For \(p=1\), using (23) and (24), we directly check that
Now assume that (32) is true for some \(p\ge 1\). First we compute that
Before we proceed we shall show that for \(p\ge 2\)
if only \(i\in \{2,\ldots ,p\}\). Indeed we have
thanks to (20). Now formulas (21) and (22) imply that
Now applying (33) if needed (that is if \(p>1\)) we conclude that
By the induction principle, using formula (27) and formula (28) (for \(i=1\)) we obtain
Finally we need to consider two cases. If p is odd, using Lemma 3.5, we obtain
If p is even, again from Lemma 3.5 we get
The proof is completed. \(\square \)
Lemma 3.7
If \(k>3\) then for every \(p\ge 1\) we have
Proof
If \(p=1\) we have
For \(p>1\) we have in general
Now the thesis follows immediately from (20) and the fact that for \(k>3\) we always have \(R(e_1,e_{k-1})e_3=0\). \(\square \)
Lemma 3.8
Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) (\(\dim M\ge 4\)) be a non-degenerate affine hypersurface with a locally equiaffine transversal vector field \(\xi \) and an almost symplectic form \(\omega \). If \(R^p\omega =0\) for some \(p\ge 1\) and
is the Jordan decomposition of S as stated in the Lemma 2.4 then \(\dim S_1\le 3\).
Proof
Let us assume that \(S_1\) has the form (15) and \(k>3\). If \(\alpha \ne 0\), from Corollary 3.2 we obtain
From Lemma 3.5, formula (30) we have
From Lemma 3.6, formula (32) we get
since \(\omega (e_2,e_k)=0\). Therefore \(\omega (e_i,e_k)=0\) for \(i\in \{1,\ldots ,2n\}\) what contradicts assumption that \(\omega \) is non-degenerate. Thus it must be \(\alpha =0\).
When \(\alpha =0\) from Lemma 3.4, formula (28) we get \(\omega (e_i,e_k)=0\) for \(i\in \{1,2,\ldots ,2n\}\setminus \{2,k\}\). Of course \(\omega (e_k,e_k)=0\) as well. From Lemma 3.6, formula (32) we have that also \(\omega (e_2,e_k)=0\), so again we obtain that \(\omega \) is degenerate thus \(\dim S_1\) cannot exceed 3. \(\square \)
Lemma 3.9
If \(S_1\) is a 3-dimensional real Jordan block, \(p\ge 1\) then
Proof
From the Gauss equation we have
Now, for \(p=1\) we easily check that
Let us assume that (36) is true for some \(p\ge 1\). Using (37) we compute
Now by the induction principle (36) holds for all \(p\ge 1\). \(\square \)
Lemma 3.10
If \(S_1\) is a 3-dimensional real Jordan block, \(p\ge 2\) and \(i,j>3\) then
Proof
Using the Gauss equetion let us note that
Now, for \(p\ge 2\) we have
since \(R(e_1,e_2)e_1=R(e_1,e_2)e_i=R(e_1,e_2)e_j=0\) and \(R(e_1,e_2)e_2=\varepsilon S_1 e_1=\varepsilon \alpha e_1+\varepsilon e_2\). In consequence, by the induction principle
for all \(p\ge 2\). We also compute
where the last equality follows from (41). The induction principle implies that (38) holds for all \(p\ge 2\). \(\square \)
Lemma 3.11
If \(S_1\) is a 3-dimensional real Jordan block, \(p\ge 1\) and \(\alpha =0\) then
for any \(i \in \{1,\ldots ,2n\}\setminus \{3\}\).
Proof
First notice that (42) holds trivially for \(i=2\). Now we assume that \(i \in \{1,\ldots ,2n\}\setminus \{2,3\}\). Note that from the Gauss equation and assumption \(\alpha =0\) we have
For \(p=1\) we easily get
Let us assume that (42) holds for some \(p\ge 1\), we shall show that it also holds for \(p+1\). Indeed we obtain
thanks to (43). Now by the induction principle (42) holds for all \(p\ge 1\). \(\square \)
Lemma 3.12
If \(S_1\) is a 3-dimensional real Jordan block, \(p\ge 1\) and \(\alpha =0\) then
Proof
First we notice that (44) is satisfied for \(p=1\). Indeed, since
we easily obtain
Now assume that (44) holds for some \(p\ge 1\), we shall show that it also holds for \(p+1\). From the Gauss equation and our assumptions we have
Now we obtain
Now by the induction principle (44) holds for all \(p\ge 1\). \(\square \)
Lemma 3.13
Let us assume that \(S_1\), \(S_2\) (from Lemma 2.4) are 2-dimensional real Jordan blocks. That is
where \(\alpha , \beta \in \mathbb {R}\). We also assume that \(H_1, H_2\) have the form
where \(\varepsilon ,\eta \in \{-1,1\}\).
Then for every \(i\in \{1,\ldots ,2n\}\setminus \{2,4\}\) we have
Proof
First note that from the Gauss equation we easily obtain that \(R(e_1,e_3)e_i =0\) for \(i\in \{1,\ldots ,2n\}\setminus \{2,4\}\). In particular
Thus (45) follows immediately.
In order to prove (46) first notice that (46) is satisfied for \(p=0\), since \(R(e_1,e_3)e_i=0\) and
Now assume that (46) holds for some \(p\ge 0\), we shall show that it also holds for \(p+1\). From the Gauss equation we also have
where the last equality follows from (45). Finally we have
By the induction principle (46) holds for all \(p\ge 0\). \(\square \)
Lemma 3.14
Let \(S_1\), \(S_2\) be 2-dimensional real Jordan blocks like in Lemma 3.13. If \(\alpha = \beta =0\) then
for every \(p\ge 1\).
Proof
First note that formulas (47), (48) and (49) from the proof of Lemma 3.13 are still valid in our case. Using them and taking into account that \(\alpha = \beta =0\) we easily compute that
and
That is (50) and (51) are true for \(p=1\). Now assume that (50) and (51) hold for some \(p\ge 1\). Again using (47)–(49) and the fact that \(\alpha = \beta =0\) we obtain
In a similar way we show that
Now by the induction principle (50) and (51) hold for all \(p\ge 1\). \(\square \)
Theorem 3.15
Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) (\(\dim M\ge 4\)) be a non-degenerate affine hypersurface with a locally equiaffine transversal vector field \(\xi \) and an almost symplectic form \(\omega \). Let
be the Jordan decomposition of S as stated in the Lemma 2.4. If \(R^p\omega =0\) for some \(p\ge 1\) then \(\dim S_1\le 2\) and \(\dim S_i=1\) for \(i=2,\ldots ,q\).
Proof
By Lemma 3.8\(\dim S_1\le 3\). If \(\dim S_1=3\), using Corollary 3.2 we obtain
By Lemma 3.9\(\omega (e_1,e_2)=0\). Now using Lemma 3.10 we obtain
for \(i,j>3\). Since \(\dim M\ge 4\) and h is non-degenerate there exist \(i,j>3\) such that \(h(e_i,e_j)\ne 0\) and in consequence \(\omega (e_1,e_3)=0\). If \(\alpha \ne 0\) then from (53) we get \(\omega (e_2,e_3)=\ldots =\omega (e_{2n},e_3)=0\), so \(\omega \) is degenerate. That is we must have \(\alpha =0\). Now using Lemmas 3.11 and 3.12 we show that \(\omega (e_2,e_i)=0\) for \(i=1,\ldots ,2n\) that is \(\omega \) is degenerate again. In consequence the case \(\dim S_1=3\) is not possible.
Assume now that \(\dim S_1=2\). If \(\dim S_2=2\) then
where \(\alpha , \beta \in \mathbb {R}\). If \(R^p\omega =0\) for some \(p\ge 1\) then also \(R^{2p+1}\omega =0\). Now using Lemma 3.13 we get
for \(i\in \{1,\ldots ,2n\}\setminus \{2,4\}\). In particular, (for \(i=1\)) we get that \(\omega (e_1,e_2)=0\). On the other hand by Corollary 3.2 we have
Note that case \(\alpha \ne 0\) is not possible since then \(\omega (e_i,e_2)=0\) for all \(i\in \{1,\ldots ,2n\}\) and \(\omega \) is degenerate. Thus we must have \(\alpha =0\). In this case Lemma 3.13 implies that \(\omega (e_i,e_2)=0\) for \(i\in \{1,\ldots ,2n\}\setminus \{2,4\}\). Since, without loss of generality, we can exchange \(S_1\) with \(S_2\) we also get that \(\beta =0\). Now by Lemma 3.14 we get that \(\omega (e_2,e_4)=0\) and \(\omega \) is degenerate, so this case is also not possible. Summarising we must have \(\dim S_1\le 2\) and \(\dim S_i=1\) for \(i=2,\ldots ,q\), what completes the proof. \(\square \)
4 Complex Jordan Blocks
In this chapter we study properties of complex Jordan blocks of the shape operator S. Before we proceed, to simplify proofs in this chapter, we need to do slight modification in the notation of Lemma 2.4.
Let \(\{e_1,\ldots ,e_{2n}\}\) be the basis of \(T_xM\) from Lemma 2.4. Without loss of generality rearranging and renaming vectors \(e_1,\ldots ,e_{2n}\) we can change order of \(S_i\) and \(H_i\) in such way that \(S_{1},\ldots ,S_{r}\) will be complex blocks and \(S_{r+1},\ldots ,S_{r+q}\) will be real blocks. If we assume that \(S_1\) is a 2k-dimensional block, \(k\ge 1\) in the new notation we will have
where \(\alpha , \beta \in \mathbb {R}\), \(\beta \ne 0\) and
Moreover, vectors \(\{e_1,\ldots ,e_{2k}\}\) will be a basis for \(S_1\).
In all the below lemmas (if not stated otherwise) we always assume that \(S_1\) and \(H_1\) are as above.
Let us start with the following three lemmas related to 2-dimensional complex Jordan blocks.
Lemma 4.1
If \(S_1\) is a 2-dimensional complex Jordan block then for every \(i\in \{3,\ldots ,2n\}\) we have
Proof
Proof easily follows from the Gauss equation and the fact that \(h(e_1,e_i)=h(e_2,e_i)=0\) for \(i>2\). \(\square \)
Lemma 4.2
If \(S_1\) is a 2-dimensional complex Jordan block then for every \(p\ge 1\), \(i\in \{3,\ldots ,2n\}\) we have
Proof
We shall prove (54). For \(p=1\), using Lemma 4.1 we compute
Assume that (54) holds for some \(p\ge 1\) we shall show that it also holds for \(p+1\). Indeed, using Lemma 4.1 we have
Now we compute that
Similarly
Thus we obtain
Now by the induction principle (54) holds for all \(p\ge 1\). The formula (55) can be shown in a similar way. \(\square \)
Lemma 4.3
If \(S_1\) is a 2-dimensional complex Jordan block then for every \(p\ge 1\), \(i,j >2\) we have
Proof
To simplify computations let us denote
where \(X,Y\in {\text {span}}\{e_1,e_2\}\), \(i,j\in \{3,\dots ,2n\}\) and \(p\ge 1\).
First we shall show that (56)–(58) are true for \(p=1\). Indeed, using Lemma 4.1, we obtain
that is
Exactly in the same way we show that
For (58) we compute
Now assume that (56)–(58) are all true for some \(p\ge 1\). We compute
since for \(l=1,\ldots ,2p\), terms \(2l-1\) and 2l cancel each other. Now we easily compute that
and
In consequence
since \(R(e_1,e_2)S e_1=(\alpha ^2+\beta ^2)e_1\) and \(R(e_1,e_2)S e_2=-(\alpha ^2+\beta ^2)e_2\). Now using assumptions (56)–(58) we obtain
In a similar way we show that
Eventually
Now by the induction principle (56)–(58) hold for all \(p\ge 1\). \(\square \)
In the next three lemmas we study properties of complex Jordan block of dimension greater than 2 in relation to other Jordan blocks from the decomposition. Thus in these lemmas, we implicitly assume that the Jordan decomposition contains more than one (not necessarily complex) block.
Lemma 4.4
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\), \(i\in \{1,\ldots ,2k-1\}\setminus \{3\}\), \(s\in \{1,\ldots ,2k\}\), \(j>2k\), \(p\ge 1\). Then
Proof
First let us notice that
and
From the above properties we obtain
thus (62) is true for \(p=1\). Moreover we have that
and
Now, assume that (62) is true for some \(p\ge 1\). Using the above formulas we easily get
Now by the induction principle (62) holds for all \(p\ge 1\). \(\square \)
Lemma 4.5
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\), \(j>2k\), \(p\ge 1\). Then
Proof
For \(p=1\), by direct computations, we obtain
Now let us assume that (63) holds for some \(p\ge 1\). Since
we compute that
Since
one may easily deduce that \(R\omega (e_3,e_{2k-1},e_1,e_j)=0\) and more general
for all \(p\ge 1\). Thus we have
where the last equality follows from (63). Now by the induction principle (63) holds for all \(p\ge 1\). \(\square \)
Lemma 4.6
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\), \(i\in \{1,\ldots ,2k\}\), \(j>2k\). Then for every \(p\ge 1\) we have
Proof
First notice that
From the above we obtain
thus (64) is true for \(p=1\). Moreover, we have
Now let us assume that (64) holds for some \(p\ge 1\). Using the above formulas we obtain
Now by the induction principle (64) holds for all \(p\ge 1\). \(\square \)
Now we can prove
Corollary 4.7
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 1\). If \(R^p\omega =0\) for some \(p\ge 1\) then for every \(X\in {\text {span}} \{e_1,\ldots ,e_{2k}\}:=V\) and \(j>2k\)
Proof
For \(k=1\) Corollary 4.7 is an immediate consequence of Lemma 4.2. For \(k\ge 2\) by Lemmas 4.4, 4.5 and 4.6 we get
Since \(\det S_1\ne 0\) and \(S_1:V\rightarrow V\) is an isomorphism (so \(\{S_1e_1,\ldots ,S_1e_{2k}\}\) generate V) we obtain that
for all \(X\in V\). \(\square \)
In the next few lemmas we study intrinsic properties of 2k-dimensional complex Jordan block for \(k\ge 2\).
Lemma 4.8
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\), \(i\in \{1,\ldots ,2k-1\}\setminus \{2\}\). Then for every \(p\ge 1\)
Proof
The Gauss equation implies that
By straightforward computations we get (66) for every \(p\ge 1\). In order to prove (67) again by the Gauss equation we have
In particular, for \(p=1\), we get
Now, assume that the formula (67) is true for some \(p\ge 1\). Then, for \(p+1\), we get
Now, by (66) and the induction principle, the formula (67) holds for every \(p\ge 1\). \(\square \)
Lemma 4.9
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\), \(p\ge 1\), \(i\in \{1,\ldots , 2k\}\setminus \{1,2k-1\}\). Then
Proof
The Gauss equation implies that
By straightforward computations we get (68) for every \(p\ge 1\). In order to prove (69) again by the Gauss equation we have
In particular, for \(p=1\), we get
Now, assume that the formula (69) is true for some \(p\ge 1\). Then, for \(p+1\), we get
Now, by (68) and the induction principle, the formula (69) holds for every \(p\ge 1\). \(\square \)
Lemma 4.10
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\), \(p\ge 1\). If
for \(j\in \{3,\ldots , 2k\}\) then
for \(i\in \{1,\ldots , p\}\)
Proof
For \(p=1\) we have
since \(\omega (e_3,e_{2k-1})=\omega (e_3,e_{2k})=0\) by assumption.
Assume that (70) holds for some \(p\ge 1\) and for all \(i\in \{1,\ldots , p\}\). Let \(i_0\in \{1,\ldots , p+1\}\). If \(i_0>1\) then we have
since \(R(e_1,e_{2k-1})e_{1}=R(e_1,e_{2k-1})e_{2k-1}=R(e_1,e_{2k-1})e_{3}=0\) and \(R(e_1,e_{2k-1})e_{2k}=-S_1e_{2k-1}=-\alpha e_{2k-1}+\beta e_{2k}\). Now by (70) we obtain that for \(i_0>1\)
If \(i_0=1\) we compute
since all terms but last are equal 0 thanks to (70). Now it is enough to show that
Indeed, by Lemma 4.8 we have
where the last equality follows from the assumption that \(\omega (e_3,e_{2k-1})=\omega (e_3,e_{2k})=0\). Summarising, we have shown that
for all \(i_0\in \{1,\ldots ,p+1\}\). Now by the induction principle (70) holds for all \(p\ge 1\). \(\square \)
Lemma 4.11
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\), \(p\ge 1\). If \(\omega (e_3,e_{2k-1})=\omega (e_3,e_{2k})=0\) then
Proof
For \(p=1\) we compute
since by assumption \(\omega (e_3,e_{2k-1})=0\). Now, assume that the formula (71) is true for some \(p\ge 1\). Then, for \(p+1\), we get
where the last equality follows from Lemma 4.8 (formula (66), \(i=1,3\)). Now, using (71) we obtain
By the induction principle (71) holds for all \(p\ge 1\).
The formula (72) can be easily obtained in a similar way using (67), (71) and the principle of induction. \(\square \)
From Lemma 4.11 we immediately get
Corollary 4.12
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\), \(p\ge 1\). If \(\omega (e_3,e_{2k-1})=\omega (e_3,e_{2k})=0\) then
Lemma 4.13
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\), \(p\ge 1\). If \(\omega (e_3,e_{2k-1})=\omega (e_3,e_{2k})=0\) then for \(i\in \{1,\ldots ,p\}\)
Proof
We directly check that
so (74) is true for \(p=1\). Now let us assume that (74) is true for some \(p\ge 1\) and for all \(i\in \{1,\ldots ,p\}\). Let \(i_0\in \{1,\ldots ,p+1\}\).
When \(i_0>1\) using the fact that \(R(e_1,e_{2k-1})e_1=R(e_1,e_{2k-1})e_{2k-1}=0\) we get
where the last equality follows from Lemma 4.10.
If \(i_0=1\), using the fact that \(R(e_{2k-1},e_{2k})e_{2k-1}=0\) we obtain
since all but first and the last two terms are equal to 0 by (74). Now Lemma 4.8 and Corollary 4.12 imply that
In consequence, using (74) we get
Now the thesis follows from the induction principle. \(\square \)
To simplify further notation let us denote:
where \(i,j\in \{1,2,3\}\) and \(p\ge 1\). We have the following lemma:
Lemma 4.14
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\), \(p\ge 1\). Then
Proof
In order to prove (76) first note that
immediately follows from Lemma 4.8, formula (66). By the Gauss equation we have
Using the above equalities we easy obtain that
To prove (77) we compute
where the last equality follows from (76). The formulas (78)–(80) we prove in a similar way like (77). To prove (81) we compute
\(\square \)
From the above lemma we obtain:
Corollary 4.15
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\), \(p\ge 1\). If
for \(j\in \{3,\ldots , 2k\}\) then
Proof
From formulas (77), (78) and assumption (82) we obtain
what proves (83) and (84). Using (83) and (84) we obtain
Moreover from formulas (79), (80) and (82) we get
Now (85) immediately follows from (81). \(\square \)
Lemma 4.16
Let \(S_1\) be a 2k-dimensional complex Jordan block, \(k\ge 2\). If \(R^p\omega =0\) for some \(p\ge 1\) then for \(i\in \{3,\ldots , 2k\}\)
Proof
From Lemma 4.8, for \(i=3,\ldots ,2k-1\) we have
In particular for \(i=2k-1\) we obtain
Thus
since \(\beta \ne 0\). Now we have
That is (86) is also true for \(i=2k\). From Lemma 4.9 for \(i\in \{3,\ldots ,2k\}\setminus \{2k-1\}\) we have
Since \(\omega (e_{2k-1},e_{2k})=0\) we also get that
That is (87) also valid for \(i=2k-1\). Now from (86) and (87), for \(i\in \{3,\ldots ,2k\}\) we get
In a similar way we compute that
Since \(\alpha ^2+\beta ^2\ne 0\) the above equations imply that
for \(i\in \{3,\ldots ,2k\}\). \(\square \)
Now we are ready to prove that the decomposition from the Lemma 2.4 cannot contain complex Jordan blocks. Namely we have the following:
Theorem 4.17
Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) (\(\dim M\ge 4\)) be a non-degenerate affine hypersurface with a locally equiaffine transversal vector field \(\xi \) and an almost symplectic form \(\omega \). If \(R^p\omega =0\) for some \(p\ge 1\) and
is the Jordan decomposition of S as stated in the Lemma 2.4 then (35) does not contain complex Jordan blocks (that is \(r=0)\).
Proof
Let \(\{e_1,\ldots ,e_{2n}\}\) be the basis of \(T_xM\) from Lemma 2.4. Without loss of generality, as described at the beginning of this section, we can change order of \(S_i\) and \(H_i\) in such way that \(S_{1},\ldots ,S_{r}\) will be complex blocks and \(S_{r+1},\ldots ,S_{r+q}\) will be real blocks. Moreover, we can assume that \(\dim S_1\ge \dim S_i\) for \(i=2,\ldots ,r\).
First assume that \(S_1\) is a complex block of dimension 2k and \(k\ge 2\). By Lemma 4.16 we have that \(\omega (e_i,e_{2k-1})=\omega (e_i,e_{2k})=0\) for \(i\in \{3,\ldots , 2k\}\). Now from Corollary 4.15 (formulas (83) and (84)) we get
and
since \(R^p\omega =0\) and \(\beta \ne 0\). In particular \(\omega (e_2,e_{2k-1})=\omega (e_1,e_{2k})\) and (85) simplify to the form
Now one can easily find explicit formula for \(A_{22}^{p}\). Namely we have
On the other hand we have \(A_{22}^{p}=0\) (since \(R^p\omega =0\)) and in consequence
From Lemma 4.13 (formula (74)) we obtain
that is
Now using (89)–(91) the above implies that
In this way we have shown that \(\omega (e_1,e_{2k-1})=0\) and in consequence also \(\omega (e_2,e_{2k-1})=0\). Hence
for \(i\in \{1,\ldots , 2k\}\). From Corollary 4.7 we also have that
for \(i>2k\), that is \(\omega \) is degenerate, what leads to contradiction and we must have \(k<2\). In this way we have shown that if Jordan decomposition of S contains some complex Jordan blocks they all must be 2-dimensional.
It remained to show that also 2-dimensional complex Jordan blocks are not possible. In order to prove it let us assume that \(S_1\) is a 2-dimensional complex block. Since \(R^p\omega =0\) then also \(R^{2p}\omega =0\) and Lemma 4.2 implies that \(\omega (e_1,e_{l})=0\) for \(l=3,\ldots ,2n\). Now observe that since \(\dim M\ge 4\) there must exist \(i_0,j_0>2\) such that \(h(e_{i_0},e_{j_0})\ne 0\) (otherwise h would be degenerate). Now from Lemma 4.3 we have
That is
since \(h(e_{i_0},e_{j_0})\ne 0\). In this way we have shown that \(\omega \) is degenerate since \(\omega (e_1,e_{l})=0\) for \(l=1,\ldots ,2n\), what leads us again to contradiction. \(\square \)
5 Main Results
Before we proceed with main results of this paper we need to recall the following two lemmas:
Lemma 5.1
([19]). Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) be a non-degenerate affine hypersurface (\(\dim M\ge 4\)) with a transversal vector field \(\xi \) and an almost symplectic form \(\omega \). Let \(x\in M\). If there exist a natural number \(2 \le s\le 2n\) and a basis \(\{e_1,\ldots ,e_{2n}\}\) of \(T_xM\) such that \(h(e_i,e_j)=\varepsilon _i\delta _{ij}\), \(\varepsilon _i=\pm 1\) for \(i,j =1,\ldots , s\), \(h(e_i,e_j)=0\) for \(i=1,\ldots , s\), \(j=s+1,\ldots , 2n\) and \(Se_i=\lambda _i e_i\) for \(i=1,\ldots , s\), \(\lambda _i\in \mathbb {R}\). Then for every \(k, j=1,2,\ldots ,s\), \(k\ne j\) and for every \(i=1,\ldots ,2n\), \(i\ne j\), \(i\ne k\) we have
for every \(l\in \mathbb {N}\).
Lemma 5.2
([19]). Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) be a non-degenerate affine hypersurface (\(\dim M\ge 4\)) with a transversal vector field \(\xi \) and an almost symplectic form \(\omega \). Let \(x\in M\) and let \(X, Y, Z_1, Z_2\) be vector fields from \(T_xM\) such that \(SX=\lambda X\), \(SZ_1=0\), \(SZ_2=0\), \(h(Z_1,Z_2)=0\) and \(h(Y,Z_2)=0\). Then, for every \(l\ge 1\) we have
We will need also below three lemmas.
Lemma 5.3
Let \(\{e_1,e_2,e_3\}\) be vectors such that \(Se_1=\alpha e_1+e_2\), \(Se_2=\alpha e_2\), \(Se_3=\lambda e_3\) and \(h(e_1,e_1)=h(e_2,e_2)=h(e_1,e_3)=h(e_2,e_3)=0\), \(h(e_1,e_2)=\eta \), \(h(e_3,e_3)=\varepsilon \), \(\eta \ne 0, \varepsilon \ne 0\). Then for every \(p\ge 1\) we have
Proof
By straightforward computations we obtain
Now for \(p\ge 1\) and \(i,j\in \{1,2\}\) we get
Using different configurations of i and j we obtain the following four relations:
Now from (98) to (99), by the induction principle we easily obtain that
for all \(p\ge 1\). In consequence from (100) to (101) we get that also
and
for all \(p\ge 2\). Now (102) simplify to the form
for \(p\ge 2\). Note that using (95)–(97) one may easily show that (103) is also true for \(p=1\). From (95) we see that (94) holds for \(p=1\). Assume now that (94) is true for some \(p\ge 1\). From (103) we compute that
Now by the induction principle (94) is true for all \(p\ge 1\). \(\square \)
Lemma 5.4
Let \(\{e_1,e_2,\ldots ,e_{2n}\}\) be vectors such that \(Se_1=e_2\), \(Se_2=0\), \(h(e_1,e_1)=h(e_2,e_2)=0\), \(h(e_1,e_2)=\eta \) and \(Se_i=\lambda _{i-2} e_i\), \(h(e_1,e_i)=h(e_2,e_i)=0\), \(h(e_i,e_i)=\varepsilon _{i-2}\) for \(i=3,\ldots ,2n\) where \(\eta \ne 0, \varepsilon \ne 0\). Then for every \(p\ge 1\) and for every \(i=3,\ldots ,2n\) we have
Proof
First we shall show that for every \(p\ge 1\) and for every \(i=3,\ldots ,2n\) the following formula holds:
For \(p=1\) we have
since
Now assume that (105) holds for some \(p\ge 1\), we shall show that it also holds for \(p+1\). From the Gauss equation we have
Now, using (106) and (107) we obtain
since all terms but first are equal 0 due to the following identities
Now by the induction principle (105) is true for all \(p\ge 1\).
Now we can prove (104). First we check that (104) is satisfied for \(p=1\) and \(p=2\). Indeed, since
by straightforward computations we easily obtain that
and
Let us assume that (104) holds for some \(p\ge 2\). Then using (108) we compute
From (109) it follows that for \(j=2,\ldots ,p\) and \(i\in \{1,\ldots ,2n\}\setminus \{2\}\)
We also have
for \(i=3,2,\ldots ,2n\) thanks to (109) and since \(p\ge 2\). Now we obtain
where the last two equalities follow from (109) to (105) respectively. By the induction principle (104) is true for all \(p\ge 1\). \(\square \)
Lemma 5.5
Let \(\{e_1,e_2,\ldots ,e_{2n}\}\) be vectors with properties like in the Lemma 5.4. Then for every \(p\ge 1\) we have
Proof
Since basis \(\{e_{1},\ldots ,e_{2n}\}\) satisfy conditions of Lemma 5.4. in particular we have (109) and (110). We also have
By direct computations we check that
It means that (111) is true for \(p=1\) and (112) is true for \(p=1,2\).
In order to show that (111) is true for all \(p\ge 1\), using (113) we compute that
since all terms but first are equal to 0 due to (109). Now by the induction principle (111) is true for all \(p\ge 1\).
To prove (112) let us assume that (112) is true for some \(p\ge 2\) we compute
Now using (110) (for \(i=1\)) we obtain
where the last equalities are consequence of (112) and (111) (for \(p-1\)). By the induction principle (112) is true for all \(p\ge 1\). \(\square \)
Theorem 5.6
Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) (\(\dim M\ge 4\)) be a non-degenerate affine hypersurface with a locally equiaffine transversal vector field \(\xi \) and an almost symplectic form \(\omega \). If \(R^p\omega =0\) for some \(p\ge 1\) then for every point \(x\in M\) either \(S=0\) in x or there exists a basis \({e_1,\ldots ,e_{2n}}\) of \(T_xM\) such that S in this basis has the form
Proof
From Theorem 4.17 we have that Jordan block decomposition of S do not contain complex blocks. From Theorem 3.15 we also know that S contains at most one real Jordan block of dimension 2 and remaining blocks are all of dimension 1.
Now (rearranging vectors \({e_1,\ldots ,e_{2n}}\) if needed) S and h can be represented in one of the following two forms:
or
where \(\varepsilon _{i}\in \{-1,1\}\) for \(i=1,\ldots ,2n\), \(\eta \in \{-1,1\}\) and
We need to show that \(\lambda _1=\cdots =\lambda _{2n}=0\) (respectively \(\alpha =0\) and \(\lambda _1=\cdots =\lambda _{2n-2}=0\)).
First assume that (115) holds. Since \(\omega \) is non-degenerate there exists \(i_0\) such that \(\omega (e_1,e_{i_0})\ne 0\). If \(i_0>2\) then using Lemma 5.1 (\(s=2n\), \(k=1\), \(j=2\), \(i=i_0\)) we get
If \(i_0=2\) then from Lemma 5.1 (\(s=2n\), \(k=1\), \(j=3\), \(i=i_0\)) we get
Since \(R^p\omega =0\) then also \(R^{2p}\omega =0\) and the above implies that
Taking into account (117) we deduce that \(\lambda _2=\lambda _3=0\) if \(i_0>2\) and \(\lambda _3=\lambda _4=0\) if \(i_0=2\). Now, if \(i_0\ne 4\) using Lemma 5.2 (\(X=e_1\), \(Y=e_{i_0}\), \(Z_1=e_3\), \(Z_4=e_4\)) we get
If \(i_0=4\) then we have that also \(\lambda _2=0\) and in this case from Lemma 5.2 (\(X=e_1\), \(Y=e_{i_0}\), \(Z_1=e_2\), \(Z_2=e_3\)) we get
The above implies that \(\lambda _1=0\) and thanks to (117) we get that \(S=0\).
Now assume that S and h have the form (116). First we shall show that \(\alpha =0\). For this purpose note that from Corollary 3.2 (\(k=2\)) we have
for \(i\in \{2,\ldots ,2n\}\). From Lemma 5.3 (\(\varepsilon =\varepsilon _1\)) we have
Since \(R^p\omega =0\) and \(\eta ,\varepsilon _1\ne 0\) we obtain
and in consequence \(\alpha =0\) (since \(\omega \) is non-degenerate).
Now we are able to show that \(\lambda _1=\cdots =\lambda _{2n-2}=0\). Indeed, from (117) it follows that it is enough to show that \(\lambda _1=0\). Since \(\alpha =0\), the basis \(\{e_{1},\ldots ,e_{2n}\}\) satisfy conditions of Lemma 5.4 and Lemma 5.5. Thus using formulas (104), (111) and (112) and taking into account that \(R^p\omega =0\) we obtain
Since \(\omega \) is non-degenerate there must exist \(i_0\in \{1,\ldots ,2n\}\setminus \{3\}\) such that \(\omega (e_{i_0},e_{3})\ne 0\). If \(i_0>3\) then from (118) we immediately get that \(\lambda _1=0\). If \(i_0=2\) then (119) implies that \(\lambda _1=0\). Eventually, if \(i_0=1\) and \(\omega (e_{2},e_{3})=0\) we obtain that \(\lambda _1=0\) from (120). The proof of the theorem is completed. \(\square \)
As a consequence of Theorem 5.6 we obtain
Theorem 5.7
Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) (\(\dim M\ge 4\)) be a non-degenerate affine hypersurface with a locally equiaffine transversal vector field \(\xi \) and an almost symplectic form \(\omega \). If \(R^k\omega =0\) for some \(k\ge 1\) then the shape operator S has the rank \(\le 1\).
Proof
From Theorem 5.6 it follows that for every \(x\in M\) either \(S_x=0\) or \(S_x\) is of the form (114) thus \({\text {rank}}S_x\le 1\) for every \(x\in M\). \(\square \)
Recall that we have the following lemma ([19]).
Lemma 5.8
([19]). Let T be a tensor of type (0, p) and let \(\nabla \) be an affine torsion-free connection. Then for every \(k\ge 1\) and for any \(2k+p\) vector fields \(X_{\pm 1}^1,\ldots ,X_{\pm 1}^k\), \(Y_1,\ldots ,Y_p\) the following identity holds:
where \(\mathcal {J}=\{a:I_k\rightarrow \{-1,1\}\}\) and \({\text {sgn}}a:=a(1)\cdot \ldots \cdot a(k)\).
From Theorem 5.7 and Lemma 5.8 we have the following
Theorem 5.9
Let \(f:M\rightarrow \mathbb {R}^{2n+1}\) (\(\dim M\ge 4\)) be a non-degenerate affine hypersurface with a locally equiaffine transversal vector field \(\xi \) and an almost symplectic form \(\omega \). If \(\nabla ^p\omega =0\) for some \(p\ge 1\) then the shape operator S has the rank \(\le 1\).
We conclude this section with the following example
Example 5.10
Let \(n\ge 2\) and let \(\gamma ,\alpha _i:\mathbb {R}\rightarrow \mathbb {R}^{2n+1}\) be curves given as follows:
for \(i=0,\ldots ,2n-3\). Let \(\varepsilon _i\in \{-1,1\}\) for \(i=1,\ldots ,2n-3\).
Now let us consider an immersion \(f:\mathbb {R}\setminus \{0\}\times (0,\infty )\times (0,\frac{\pi }{2})^{2n-2}\rightarrow \mathbb {R}^{2n+1}\) given by the formula:
together with the transversal vector field
By straightforward computations we get
and
Thus f is a non-degenerate equiaffine hypersurface with the second fundamental form of signature \((1,-1,\varepsilon _0,\varepsilon _{1},\ldots ,\varepsilon _{2n-3})\) where \(\varepsilon _0=1\) for \(x>0\) and \(\varepsilon _0=-1\) for \(x<0\). Note also that \(R\ne 0\) since \(S\ne 0\).
Now let us define
such that \(\omega _{i,j}=-\omega _{j,i}\) and \(\det \Omega \ne 0\). That is \(\Omega \) is a symplectic form. We easily check that
and
thus \(R\Omega \ne 0\) and \(R^2\Omega \ne 0\) if only \(\omega _{1,2}\ne 0, \omega _{2,3}\ne 0\). On the other hand one may show that \(R^3\Omega = 0\).
Data Availability Statement
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
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Szancer, M., Szancer, Z. Affine Hypersurfaces of Arbitrary Signature with an Almost Symplectic Form. Results Math 78, 67 (2023). https://doi.org/10.1007/s00025-023-01838-1
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DOI: https://doi.org/10.1007/s00025-023-01838-1