Isometric Embeddings of Hyperkähler $\mathrm{T}^*\mathbb{CP}^{n}$

Abstract

We construct explicit $U(n)$-equivariant isometric embeddings $T^*\mathbb{CP}^{n-1} \hookrightarrow \mathbb{R}^{n^4+2n^2-1}$, $n\ge 2$, with respect to Calabi's hyperkähler metric on $T^*\mathbb{CP}^{n-1}$. To achieve this, we first construct a diffeomorphism from $T^*\mathbb{CP}^{n-1}$ onto the manifold of rank-1 projections in $M_n(\mathbb{C})$, such that the pullback of half of the trace-class norm is a Kähler potential for the hyperkähler metric. We then use the Hilbert space structure of $M_n(\mathbb{C})$ to produce the isometric embeddings into Euclidean space.