Abstract
This paper presents two results in the realm of conformal Kaehler submanifolds. These are conformal immersions of Kaehler manifolds into the standard flat Euclidean space. The proofs are obtained by making a rather strong use of several facts and techniques developed in Chion and Dajczer (Proc Edinb Math Soc 66:810–833, 2023) for the study of isometric immersions of Kaehler manifolds into the standard hyperbolic space.
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Let \(f:M^{2n}\rightarrow {\mathbb {R}}^{2n+p}\) denote a conformal Kaehler submanifold. Thus \((M^{2n},J)\) is a Kaehler manifold of complex dimension \(n\ge 2\) and f a conformal immersion into Euclidean space that lies in codimension p. Thus, there is a positive function \(\lambda \in C^\infty (M)\) such that the Kaehler metric and the one induced by f relate by \({\langle },\,{\rangle }_f=\lambda ^2{\langle },\,{\rangle }_{M^{2n}}\).
Conformal Kaehler submanifolds laying in the low codimensions \(p=1\) and \(p=2\) have already been considered in [1]. In this paper, we are interested in higher codimensions although not too large in comparison to the dimension of the manifold.
Our first result, provides a necessary condition for the existence of a conformal immersion in codimension at most \(n-3\) in terms of the sectional curvature of the Kaehler manifold.
Let \(f:M^{2n}\rightarrow {\mathbb {R}}^{2n+p}\), \(p\le n-3\), be a conformal Kaehler submanifold. Then at any \(x\in M^{2n}\) there is a complex vector subspace \(V^{2m}\subset T_xM\) with \(m\ge n-p\) such that the sectional curvature of \(M^{2n}\) satisfies \(K_M(S,JS)(x)\le 0\) for any \(S\in V^{2m}\).
Notice that the conclusion of Theorem 1 remains valid if the Euclidean ambient space \({\mathbb {R}}^{2n+p}\) is replaced by any locally conformally flat manifold of the same dimension.
Our second result characterizes a submanifold, in terms of the degree of positiveness of the sectional curvature, as being locally the Example 2 presented below.
The light-cone \({\mathbb {V}}^{m+1}\subset {\mathbb {L}}^{m+2}\) of the standard flat Lorentzian space is any one of the two connected components of the set of all light-like vectors, namely,
endowed with the (degenerate) induced metric. The Euclidean space \({\mathbb {R}}^m\) can be realized as an umbilic hypersurface of \({\mathbb {V}}^{m+1}\) as follows: Take vectors \(v,w\in {\mathbb {V}}^{m+1}\) such that \({\langle }v,w{\rangle }=1\) and a linear isometry \(C:{\mathbb {R}}^m\rightarrow \{v,w\}^\perp \). Let \(\psi :{\mathbb {R}}^m\rightarrow {\mathbb {V}}^{m+1}\subset {\mathbb {L}}^{m+2}\) be defined by
Then \(\psi \) is an isometric embedding of \({\mathbb {R}}^m\) as an umbilical hypersurface in the light cone which is the intersection of \({\mathbb {V}}^{m+1}\) with an affine hyperplane. Namely, we have that
The normal bundle of \(\psi \) is \(N_\psi {\mathbb {R}}^m=\hbox {span\,}\{\psi ,w\}\) and its second fundamental form is
Proposition 9.9 in [3] gives an elementary correspondence between the conformal immersions in Euclidean space and the isometric immersions into the light cone which goes as follows: Associated to a given conformal immersion \(f:M^m\rightarrow {\mathbb {R}}^{m+p}\) with conformal factor \(\lambda \in C^\infty (M)\) there is the associated isometric immersion defined by
Conversely, any isometric immersion \(F:M^m\rightarrow {\mathbb {V}}^{m+p+1}{\setminus }{\mathbb {R}}w\subset {\mathbb {L}}^{m+p+2}\) gives rise to an associated conformal immersion \(f:M^m\rightarrow {\mathbb {R}}^{m+p}\) given by \(\psi \circ f=\pi \circ F\) with conformal factor \(1/{\langle }F,w{\rangle }\). Here \(\pi :{\mathbb {V}}^{m+p+1}\setminus {\mathbb {R}}w\rightarrow {\mathbb {R}}^{m+p}\) is the projection \(\pi (x)=x/{\langle }x,w{\rangle }\).
. Let the Kaehler manifold \(M^{2n}\) be the Riemannian product of one hyperbolic plane and a set of two-dimensional round spheres such that
If \(f_1\) is the inclusion \({\mathbb {H}}^2_c\subset {\mathbb {L}}^3\) and \(f_2:{\mathbb {S}}^2_{c_2}\times \cdots \times {\mathbb {S}}^2_{c_n}\rightarrow {\mathbb {S}}^{3n-4}_c \subset {\mathbb {R}}^{3n-3}\) is the product of umbilical spheres then the map \(\psi ^{-1}\circ (f_1\times f_2):M^{2n}\rightarrow {\mathbb {R}}^{3n-2}\) is a conformal Kaehler submanifold.
FormalPara Theorem 3. Let \(f:M^{2n}\rightarrow {\mathbb {R}}^{2n+p}\), \(2\le p\le n-2\), be a connected conformal Kaehler submanifold. Assume that at a point \(x_0\in M^{2n}\) there is a complex tangent vector subspace \(V^{2m}\subset T_{x_0}M\) with \(m\ge p+1\) such that the sectional curvature of \(M^{2n}\) satisfies \(K_M(S,JS)(x)>0\) for any \(0\ne S\in V^{2m}\). Then \(p=n-2\) and f(M) is an open subset of the submanifold given by Example 2.
1 The Proofs
Let \(V^{2n}\) and \({\mathbb {L}}^p\), \(p\ge 2\), be real vector spaces such that there is \(J\in Aut(V)\) which satisfies \(J^2=-I\) and \({\mathbb {L}}^p\) is endowed with a Lorentzian inner product \({\langle },\,{\rangle }\). Then let \(W^{p,p}={\mathbb {L}}^p\oplus {\mathbb {L}}^p\) be endowed with the inner product of signature (p, p) defined by
A vector subspace \(L\subset W^{p,p}\) is called degenerate if \(L\cap L^\perp \ne 0\).
Let \(\alpha :V^{2n}\times V^{2n}\rightarrow {\mathbb {L}}^p\) be a symmetric bilinear form and \(\beta :V^{2n}\times V^{2n}\rightarrow W^{p,p}\) the associated bilinear form given by
We have that if \(\beta (X,Y)=(\xi ,\eta )\) then
We denote the vector subspace of \(W^{p,p}\) generated by \(\beta \) by
and say that \(\beta \) is surjective if \({\mathcal {S}}(\beta )=W^{p,p}\). The (right) kernel \(\beta \) is defined by
A vector \(X\in V^{2n}\) is called a (left) regular element of \(\beta \) if \(\dim B_X(V)=r\) where \(r=\max \{\dim B_X(V):X\in V\}\) and \(B_X:V\rightarrow W^{p,p}\) is the linear transformation defined by \(B_XY=\beta (X,Y)\). The set \(RE(\beta )\) of regular elements of \(\beta \) is easily seen to be an open dense subset of \(V^{2n}\), for instance see Proposition 4.4 in [3].
It is said that \(\beta \) is flat if it satisfies that
If \(\beta \) is flat and \(X\in RE(\beta )\) we have from Proposition 4.6 in [3] that
Proposition 4
Let \(\beta :V^{2n}\times V^{2n}\rightarrow W^{p,p}\), \(p\le n\), be flat and surjective. Then
Proof
This is condition (9) in Proposition 11 of [2]. \(\square \)
Let \(V^{2n}\) be endowed with a positive definite inner product \((,\,)\) with respect to which \(J\in Aut(V)\) is an isometry. Assume that there is a light-like vector \(w\in {\mathbb {L}}^p\) such that
Let \(U_0^s\subset {\mathbb {L}}^p\) be the s-dimensional vector subspace given by
where \(\pi _1:W^{p,p}\rightarrow {\mathbb {L}}^p\) denotes the projection onto the first component of \(W^{p,p}\). From Proposition 9 in [2] we know that
In addition, if \({{{\mathcal {S}}}}(\beta )\) is a degenerate vector subspace then \(1\le s\le p-1\) and there is a light-like vector \(v\in U_0^s\) such that
Proposition 5
Let the bilinear form \(\beta :V^{2n}\times V^{2n}\rightarrow W^{p,p}\) be flat and the vector subspace \({{{\mathcal {S}}}}(\beta )\) degenerate. Then \(L=\hbox {span\,}\{v,w\}\subset {\mathbb {L}}^p\) is a Lorentzian plane. Moreover, choosing v such that \({\langle }v,w{\rangle }=-1\) and setting \(\beta _1=\pi _{L^\perp \times L^\perp }\circ \beta \), we have
Furthermore, if \(s\le n\) then
Proof
We obtain from (6) that
From (4) and the fact that J is an isometry with respect to \((,\,)\), we also have that
In particular, we have \({\langle \!\langle }\beta (X,X),(w,0){\rangle \!\rangle }<0\) for any \(0\ne X\in V^{2n}\), which jointly with (9) implies that v and w are linearly independent and thus span a Lorentzian plane.
Since w is light-like and v satisfies \({\langle }v,w{\rangle }=-1\), we have
where \(\alpha _{L^\perp }\) denotes the \(L^\perp \)-component of \(\alpha \). Then (9) and (10) yield
and
from which we obtain (7).
We have from (5) and (6) that \(w\notin U_0^s+L^\perp \). Hence \(\dim (U_0^s+L^\perp )= p-1\). It then follows from
that \(U_1=U_0^s\cap L^\perp \) satisfies
and we have from (5), (6) and (7) that \({{{\mathcal {S}}}}(\beta _1)=U_1^{s-1}\oplus U_1^{s-1}\).
From (7) we obtain that
and hence also the bilinear form \(\beta _1:V^{2n}\times V^{2n}\rightarrow L^\perp \oplus L^\perp \) is flat. Let \(X\in RE(\beta _1)\) and set \(N_1(X)=\ker B_{1X}\) where \(B_{1X}Y=\beta _1(X,Y)\). To obtain (8) it suffices to show that \(N_1(X)={\mathcal {N}}(\beta _1)\) since then \(\dim {\mathcal {N}}(\beta _1) =\dim N_1(X)\ge 2n-2\dim U_1= 2n-2\,s+2\).
If \(\beta _1(Y,Z)=(\xi ,\eta )\) then by (2) and (7) we have \(\beta _1(Z,Y)=(\xi ,-\eta )\). If \(Y,Z\in N_1(X)\) it follows from (3) that
Hence \(\beta _1|_{N_1(X)\times N_1(X)}=0\) since the inner product induced on \(U_1^{s-1}\) is positive definite. Now let \(\beta _1(Y,Z)=(\delta ,\zeta )\) where \(Y\in V^{2n}\) and \(Z\in N_1(X)\). Then the flatness of \(\beta _1\) yields
and therefore \(\beta _1|_{V\times N_1(X)}=0\). \(\square \)
Proposition 6
Let the bilinear form \(\beta :V^{2n}\times V^{2n}\rightarrow W^{p,p}\) be flat and satisfy
where \(\gamma :V^{2n}\times V^{2n}\rightarrow W^{p,p}\) is the bilinear form defined by
If the vector subspace \({{{\mathcal {S}}}}(\beta )\) is degenerate and \(s\le n-1\) then there is a J-invariant vector subspace \(P^{2m}\subset V^{2n}\), \(m\ge n-s+1\), such that
Proof
Let \(v\in U_0^s\) be given by (6). We claim that
Since \(s\le n-1\) then (8) gives \(\dim {\mathcal {N}}(\beta _1)\ge 4\). Hence (7) yields \(\beta (S,S)=2((S,S)v,0)\) for any \(S\in {\mathcal {N}}(\beta _1)\). Thus
for any \(S\in {\mathcal {N}}(\beta _1)\) of unit length. On the other hand, we obtain from (2) and (7) that \(\beta (S,Y)=\beta (Y,S)=0\) for any \(S\in {\mathcal {N}}(\beta _1)\) and \(Y\in \{S,JS\}^\perp \). Then (12) and (14) give \({\langle }\alpha (X,Y),v{\rangle }=0\) for any \(X\in V^{2n}\) and \(Y\in \{S,JS\}^\perp \) where \(S\in {\mathcal {N}}(\beta _1)\). Now that \(\dim {\mathcal {N}}(\beta _1)\ge 4\) yields the claim.
Choosing \(v\in U_0^s\) as in Proposition 5 it follows from (4) and (13) that
Then we obtain from (15) that
Set \(P^{2m}={\mathcal {N}}(\beta _1)\) where \(2m=\dim {\mathcal {N}}(\beta _1)\ge 2n-2s+2\) by (8). From (7) we have \(\beta (Z,S)=2((Z,S)v,(Z,JS)v)\) for any \(S\in P^{2m}\) and \(Z\in V^{2n}\). Then (12) gives
for any \(S\in P^{2m}\) and \(X,Y,Z\in V^{2n}\). Hence \({{{\mathcal {S}}}}(\gamma |_{V\times P})\) and \({{{\mathcal {S}}}}(\beta )\) are orthogonal vector subspaces. From (11) we have
Then by (5) the vector subspaces \({{{\mathcal {S}}}}(\gamma |_{V\times P})\) and \(U_1^{s-1}\oplus U_1^{s-1}\) are orthogonal, and thus
for any \(X\in V^{2n}\), \(S\in P^{2m}\) and \(\xi \in U^{s-1}_1\). Since \(U_1^{s-1}\subset L^\perp \) then
Let \({\mathbb {L}}^p=U_1^{s-1}\oplus U_2^{p-s-1}\oplus L\) be an orthogonal decomposition. Then (5) and (16) give
for any \(X,Y\in V^{2n}\) and \(\xi _2\in U_2^{p-s-1}\). Thus
Having \(U_2^{p-s-1}\) a positive definite induced inner product, we obtain from (15), (17) and (18) that
for any \(S\in P^{2m}\). \(\square \)
Given a conformal immersion \(f:M^{2n}\rightarrow {\mathbb {R}}^{2n+p}\) with conformal factor \(\lambda \in C^\infty (M)\) we have the associated isometric immersion \(F=\frac{1}{\lambda }\psi \circ f:M^{2n} \rightarrow {\mathbb {V}}^{2n+p+1}\subset {\mathbb {L}}^{2n+p+2}\) where \(\psi \) is given by (1). Differentiating \({\langle }F,F{\rangle }=0\) once gives \(F\in \Gamma (N_FM)\) and twice yields that the second fundamental form \(\alpha ^F:TM\times TM\rightarrow N_FM\) of F satisfies
Since \(\psi _*N_fM\subset N_FM\) the normal bundle of F decomposes as \(N_FM=\psi _*N_fM\oplus L^2\equiv {\mathbb {L}}^{p+2}\) where \(L^2\) is the Lorentzian plane subbundle orthogonal to \(\psi _*N_fM\) such that \(F\in \Gamma (L^2)\).
Let the bilinear forms \(\gamma ,\beta :T_xM\times T_xM\rightarrow N_FM(x)\oplus N_FM(x)\) be defined by
and
Proposition 7
Let \(N_FM(x)\oplus N_FM(x)\) be endowed with the inner product defined by
Then the bilinear form \(\beta \) is flat and
Proof
The proof is straightforward using that \(\beta (X,JY)=-\beta (JX,Y)\), that the curvature tensor satisfies \(R(X,Y)JZ=JR(X,Y)Z\) for any \(X,Y,Z\in T_xM\) and the Gauss equation for f; for details see the proof of Proposition 16 in [2]
\(\square \)
Proof of Theorem 1
It suffices to show that the vector subspace \({{{\mathcal {S}}}}(\beta )\) is degenerate since then the proof follows from the Gauss equation jointly with Propositions 6 and 7. If \({{{\mathcal {S}}}}(\beta )\) is not degenerate, and since we have the result given by Proposition 7, then Proposition 4 yields \(\dim {\mathcal {N}}(\beta )\ge 2n-2p-4>0\). But this is a contradiction since from (19) we have that \({\mathcal {N}}(\beta )=0\). \(\square \)
Proposition 8
Let the bilinear form \(\beta :V^{2n}\times V^{2n}\rightarrow W^{p,p}\), \(s\le n\), be flat. Assume that the vector subspace \({{{\mathcal {S}}}}(\beta )\) is nondegenerate and that (12) holds. For \(p\ge 4\) assume further that there is no non-trivial J-invariant vector subspace \(V_1\subset V^{2n}\) such that the subspace \({{{\mathcal {S}}}}(\beta |_{V_1\times V_1})\) is degenerate and \(\dim {{{\mathcal {S}}}}(\beta |_{V_1\times V_1})\le \dim V_1-2\). Then \(s=n\) and there is an orthogonal basis \(\{X_i,JX_i\}_{1\le i\le n}\) of \(V^{2n}\) such that:
-
(i)
\(\beta (Y_i,Y_j)=0\;\hbox {if}\; i\ne j \;\hbox {and}\;Y_k\in \hbox {span\,}\{X_k,JX_k\}\) for k=i,j.
-
(ii)
The vectors \(\{\beta (X_j,X_j),\beta (X_j,JX_j)\}_{1\le j\le n}\) form an orthonormal basis of \({{{\mathcal {S}}}}(\beta )\).
Proof
It follows from Proposition 15 in [2]. \(\square \)
Proof of Theorem 3
Theorem 1 gives that \(p=n-2\). In an open neighborhood U of \(x_0\) in \(M^{2n}\) there is a complex vector subbundle \({\bar{V}}\subset TM\) such that \({\bar{V}}(x_0)=V^{2m}\) and \(K_M(S,JS)>0\) for any \(0\ne S\in {\bar{V}}\). At any point of U the vector subspace \({{{\mathcal {S}}}}(\beta )\) is nondegenerate. In fact, if otherwise then by Proposition 6 there is a point \(y\in U\) and a complex vector subspace \(P^{2\ell }\subset T_yM\) with \(\ell \ge 2\) such that the sectional curvature satisfies \(K_M(S,JS)\le 0\) for any \(0\ne S\in P^{2\ell }\), in contradiction with our assumption.
By Proposition 8, there is at any \(y\in U\) an orthogonal basis \(\{X_j,JX_j\}_{1\le j\le n}\) of \(T_yM\) such that both parts hold. By part (ii) the vectors \((\xi _j,0)=\beta (X_j,X_j)\in N_FM(y)\), \(1\le j\le n\), are orthonormal. Then the argument used for the proof of Lemma 18 in [2] gives that \(F|_U\) has flat normal bundle, that \(\hbox {rank }A_{\xi _j}\)=2 for \(1\le j\le n\) and that the normal vector fields \(\xi _1,\ldots ,\xi _n\) are smooth on connected components of an open dense subset of U. Moreover, we obtain from the Codazzi equation and the use of the de Rham theorem that \(M^{2n}\) is locally a Riemannian product of surfaces \(M_1^2\times \cdots \times M_n^2\).
Having that the codimension is \(n=p+2\) and that \(\alpha ^F(Y_i,Y_j)=0\) if \(Y_i\in (E_i)\) and \(Y_j\in (E_j)\), \(i\ne j\), then by Theorem 8.7 in [3] there are isometric immersions \(g_1:M_1^2\rightarrow {\mathbb {L}}^3\) and \(g_j:M_j^2\rightarrow {\mathbb {R}}^3\), \(2\le j\le n\), such that
Since \(F(M)\subset {\mathbb {V}}^{3n-1}\subset {\mathbb {L}}^{3n}\) then \({\langle }F,F{\rangle }=0\). Hence \({\langle }g_j{}_*X_j,g_j{\rangle }={\langle }g_*X_j,g_j{\rangle }=0\) and thus \(\Vert g_j\Vert =r_j\) with \(-r_1^2+\sum _{j=2}^nr_j^2=0\). This gives that \(F(U)\subset {\mathbb {H}}^2_{c_1}\times {\mathbb {S}}^2_{c_2}\times \cdots \times {\mathbb {S}}^n_{c_n}\) where \(1/c_i=r_i^2\) and, by continuity, this also holds for F(M).\(\square \)
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References
de Carvalho, A., Chion, S., Dajczer, M.: Conformal Kaehler Euclidean submanifolds. Differ. Geom. Appl. 82, 101893 (2022)
Chion, S., Dajczer, M.: Kaehler submanifolds of the real hyperbolic space. Proc. Edinb. Math. Soc. 66, 810–833 (2023)
Dajczer, M., Tojeiro, R.: Submanifold Theory Beyond an Introduction. Series: Universitext. Springer, Berlin (2019)
Funding
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Luis J. Alías and Marcos Dajczer are partially supported by the Grant PID2021-124157NB-I00 funded by MCIN/AEI/10.13039/501100011033/ ‘ERDF A way of making Europe’, Spain, and are also supported by Comunidad Autónoma de la Región de Murcia, Spain, within the framework of the Regional Programme in Promotion of the Scientific and Technical Research (Action Plan 2022), by Fundación Séneca, Regional Agency of Science and Technology, REF, 21899/PI/22.
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Alías, L.J., Chion, S. & Dajczer, M. Conformal Kaehler Submanifolds. Results Math 79, 170 (2024). https://doi.org/10.1007/s00025-024-02203-6
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DOI: https://doi.org/10.1007/s00025-024-02203-6