The Möbius geometry of hypersurfaces

A global characterization of tubed surfaces in $\mathbb {C}^2$

“…In fact, this result holds locally. That is, if Q r,p ≡ 0 but M is not complete, then M is contained in a Hermitian quadric [3,5]. The condition Q r,p ≡ ε L r,p when ε ∈ C with |ε| = 0, 1 is also locally well understood.…”

“…We briefly recall some facts from [4] that are also relevant to the statement and proof of Theorem 1.…”

“…The proof of Theorem 1 uses the structural equations for a hypersurface in a manner similar to how they were used to prove the main results in [3,4]. In this case, the equations are used to show that the rank of the second fundamental form of M is at most one.…”

“…These vectors form an orthonormal basis of tangent vectors in C 2 along M . With these choices, it follows from Proposition 3 in [4] that both The rows and columns correspond with tangent vectors JN, X, JX, respectively, as defined in (2). This means, in particular, that the Weingarten map is given by…”

“…The following lemma was proved in [3] and is useful for computing the connection along M expressed in terms of the system (2). …”