Immersion in \(\mathbb {R}^n\) by complex spinors (Q1659528)

This article is concerned with isometric immersions of simply-connected, Riemannian spin\(^c\)-manifolds into Euclidean space. Its main theorem is a direct generalization of the corresponding theorem for spin-manifolds from [\textit{P. Bayard} et al., Pac. J. Math. 291, No. 1, 51--80 (2017; Zbl 1373.53073)]. The main theorem can be paraphrased as follows. Let \(M\) be a simply-connected Riemannian \(n\)-manifold, \(E \to M\) a vector bundle of rank \(m\), and assume that \(TM\) and \(E\) are oriented and spin\(^c\). Furthermore, let \(B: TM \times TM \to E\) be symmetric and bilinear. Then the following are equivalent: {\parindent=6mm \begin{itemize}\item[(1)] There exists a section \(\varphi\) of some spin\(^c\)-Clifford bundle (which is concretely constructed in the article) such that \[ \nabla_X \varphi = -\frac{1}{2} \sum_i e_i\cdot B(X,e_i)\cdot \varphi + \frac{1}{2}i A(X) \cdot \varphi, \quad \forall X\in TM, \] where \(iA: TP_{S^1} \to i\mathbb{R}\) is a connection that one constructs (the circle \(S^1\) comes from the spin\(^c\)-structures on \(TM\) and \(E\)). \item[(2)] There exists an isometric immersion \(F: M \to \mathbb{R}^{n+m}\) with normal bundle \(E\) and second fundamental form \(B\). \end{itemize}} The proof of \((1) \Rightarrow (2)\) proceeds in the following way: We define an \(\mathbb{R}^{n+m}\)-valued \(1\)-form \(\xi\) by \(\xi(X) := \langle\langle X \cdot \varphi,\varphi\rangle\rangle,\) where \(\langle\langle -,-\rangle\rangle\) is an inner product constructed in the article. Using \((1)\) one then proves that \(\xi\) is closed, and hence, because \(M\) is simply-connected, that there is an \(F: M \to \mathbb{R}^{n+m}\) with \(dF = \xi\). This \(F\) is the sought isometric immersion.