CR Embeddings and Kähler Manifolds with Pseudo-Conformally Flat Curvature Tensors

Kähler submanifolds and the Umehara algebra

“…This is the same as assuming that h 1 , h 2 are linearly dependent. So there is a function h(z) and complex numbers m 1 , m 2 such that h 1 = m 1 h and h 2 = m 2 h. Plugging this into equation (9) we get (1 + |k 1 (z)| 2 + |k 2 (z)| 2 ) 2 = (1 + |mh(z)| 2 ) 3 where m = |m 1 | 2 + |m 2 | 2 . Changing the variable z with z = h −1 (w) we get (1 + | k 1 (w)| 2 + | k 2 (w)| 2 ) 2 = (1 + |mw| 2 ) 3 a further change of variables w = z m gives (1 + | k 1 (z)| 2 + | k 2 (z)| 2 ) 2 = (1 + |z| 2 ) 3 but this is impossible.…”

“…This is the same as assuming that h 1 , h 2 are linearly dependent. So there is a function h(z) and complex numbers m 1 , m 2 such that h 1 = m 1 h and h 2 = m 2 h. Plugging this into equation (9) we get Case deg(P) = 2. This implies that there exist Q ∈ C[X, Y ], deg(Q) = 2, such that Q divides the four polynomials P 6 , P 7 , P 8 , P 9 .…”

“…Starting with Bochner's paper [1] such questions have been studied extensively by many authors e.g. [2,3,6,9,10,14,15,17,19]. In his PhD.…”