AI Summary & Key Takeaways
AI InsightIf a curve 𝛼 is a horizontal geodesic on 𝑀 with respect to both ∇ and ˜∇, then the dimension of horizontal distribution is equal to 2 or the Lee vector field 𝐵 of (𝑀, 𝐽, 𝑔) is vertical.
📌 3 Key Takeaways
Abstract
We noticed a small error in the proof of Theorem 3.5 in our paper cited in the heading. Here, we explicitly explain some details. The mentioned theorem in the proof asserts that the Lee vector field of the total manifold of a locally conformal Kaehler submersion cannot be vertical. But this is not true in general. Therefore, that theorem is not valid, and unfortunately, that theorem slightly affects the validity of Theorem 3.5 of our paper Pirinççi et al. (2023). We would like to update this theorem and its proof as follows: Theorem 3.5. Let 𝜋 : (𝑀, 𝐽, 𝑔) → (𝑁, 𝐽′, 𝑔′) be a l.c.K. submersion with connected fibers. If a curve 𝛼 is a horizontal geodesic on 𝑀 with respect to both ∇ and ˜∇, then the dimension of horizontal distribution is equal to 2 or the Lee vector field 𝐵 of (𝑀, 𝐽, 𝑔) is vertical.