Holomorphicity of real Kaehler submanifolds

Carvalho, Alcides de; Chion, Sergio; Dajczer, Marcos

doi:10.2422/2036-2145.202105_013

Cited by 5 publications

(16 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof: Proposition 7 in [2] gives (29). The remaining of the statement is Proposition 2.6 in [4] as well as Lemma 7 in [14].…”

Section: The Proofsmentioning

confidence: 96%

The second fundamental form of the real Kaehler submanifolds

Chión¹,

Dajczer

2023

Preprint

Self Cite

View full text Add to dashboard Cite

Let $f\colon M^{2n}\to\R^{2n+p}$, $2\leq p\leq n-1$, be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng conjectured in \cite{YZ} that if the codimension is $p\leq 11$ then, along any connected component of an open dense subset of $M^{2n}$, the submanifold is as follows: it is either foliated by holomorphic submanifolds of dimension at least $2n-2p$ with tangent spaces inthe kernel of the second fundamental form whose images are open subsets of affine vector subspaces, or it is embedded holomorphically in a Kaehler submanifold of $\R^{2n+p}$ of larger dimension than $2n$. This bold conjecture was proved by Dajczer and Gromoll just for codimension three and then by Yan and Zheng for codimension four. In this paper we prove that the second fundamental form of the submanifold behaves pointwise as expected in case that the conjecture is true. This result is a first fundamental step fora possible classification of the non-holomorphic Kaehler submanifolds lying with low codimension in Euclidean space. A counterexample shows that our proof does not work for higher codimension, indicating that proposing $p=11$ in the conjecture as the largest codimension is appropriate.

show abstract

“…Proof: Proposition 7 in [2] gives (29). The remaining of the statement is Proposition 2.6 in [4] as well as Lemma 7 in [14].…”

Section: The Proofsmentioning

confidence: 96%

The second fundamental form of the real Kaehler submanifolds

Chión¹,

Dajczer

2023

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Thus J N is constant along the fibers of Λ. Then J N is parallel with respect to the Levi-Civita connection ∇ on N 2n+2ℓ since by (3) we have that…”

Section: ) Lambdamentioning

confidence: 98%

“…Moreover, if the codimension is even then f , when restricted to any open subset of M 2n , is not allowed to be holomorphic in R 2n+2q ∼ = C n+q . Notice that conditions that yield f to be holomorphic have been given in [3] and are just minimal in [2].…”

mentioning

confidence: 99%

Real Kaehler submanifolds in codimension up to four

Chion¹,

Dajczer²

2022

Preprint

View full text Add to dashboard Cite

Let f : M 2n → R 2n+4 be an isometric immersion of a Kaehler manifold of complex dimension n ≥ 5 into Euclidean space with complex rank at least 5 everywhere. Our main result is that, along each connected component of an open dense subset of M 2n , either f is holomorphic in R 2n+4 ∼ = C n+2 or it is in a unique way a composition f = F • h of isometric immersions. In the latter case, we have that h : M 2n → N 2n+2 is holomorphic and F : N 2n+2 → R 2n+4 belongs to the class, by now quite well understood, of non-holomorphic Kaehler submanifold in codimension two. Moreover, the submanifold F is minimal if and only if f is minimal.

show abstract

“…Proof: For p ≤ 5 the proof of Proposition 10 in [1], where the inner product on U p is positive definite and α satisfies a certain condition, can be adapted to this case. With the notations in there and using the same type of arguments used there it is easy to conclude that the only cases one needs to consider are 2 ≤ κ < τ ≤ p − 1 where κ and τ are even.…”

Section: Flat Bilinear Formsmentioning

confidence: 99%

“…We cannot have B Y j (N) = U(X), j = 1, 2, since otherwise dim U(Y j ) ≤ 3 < τ . Therefore, by Fact 12 in [1] it remains to consider the case dim…”

Section: Flat Bilinear Formsmentioning

confidence: 99%

Conformal Kaehler Euclidean submanifolds

de Carvalho,

Chion,

Dajczer

2019

Preprint

View full text Add to dashboard Cite

Let f : M 2n → R 2n+ℓ , n ≥ 5, denote a conformal immersion into Euclidean space with codimension ℓ of a Kaehler manifold of complex dimension n and free of flat points. For codimensions ℓ = 1, 2 we show that such a submanifold can always be locally obtained in a rather simple way, namely, from an isometric immersion of the Kaehler manifold M 2n into either R 2n+1 or R 2n+2 , the latter being a class of submanifolds already extensively studied.

show abstract

Holomorphicity of real Kaehler submanifolds

Cited by 5 publications

References 0 publications

The second fundamental form of the real Kaehler submanifolds

The second fundamental form of the real Kaehler submanifolds

Real Kaehler submanifolds in codimension up to four

Conformal Kaehler Euclidean submanifolds

Contact Info

Product

Resources

About