Isometries of a product of Riemannian manifolds

Theorem. Let \((A, g_A)\) and \((B, g_B)\) be two compact Riemannian manifolds with irreducible holonomy groups. Let \((X, g):= (A \times B, g_A \oplus g_B)\). Then

\[ \operatorname{Isom}(X) = \begin{cases} \operatorname{Isom}(A) \times \operatorname{Isom}(B), \ \text{ if } A \text{ is not isometric to } B \\ \left ( \operatorname{Isom}(A) \times \operatorname{Isom}(B) \right ) \rtimes \mathbb{Z}/2\mathbb{Z}, \ \text{ otherwise } \end{cases} \]

This result seems to be a folklore, probably well known to the specialists, although it is hard to find it in the literature. The only discussion which I managed to find on Mathoverflow contains …

Chern connections in the context of CR-geometry

This post is a result of several discussions with Rodion Déev.

1. Preliminaries: Chern’s theorem

Fix a complex manifold \(X\) and a complex vector bundle \(E\) over \(X\). Recall that a structure of holomorphic bundle on \(E\) is given by an operator

\[\overline{\partial}_{\mathcal{E}} \colon \Gamma (E) \to \Gamma(E \otimes \Lambda^{0,1}X)\]

which satisfies \(\overline{\partial}\)-Leibniz identity

\[\overline{\partial}_{\mathcal{E}}(fs) = f\overline{\partial}_{\mathcal{E}}(s) + \overline{\partial}(f) \otimes s\]

and the integrability condition

\[\overline{\partial}_{\mathcal{E}}^2 = 0.\]

For the …

A shortcut in Kapovich’s proof of Haupt’s theorem

The Teichmüller space \(T_g\) of genus \(g\) curves carries the Hodge bundle \(\Omega T_g\), the total space of which maps into the first cohomology space \(V = H^1(S_g,\mathbb{C})\) via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image \(per(\Omega T_g) \subset V\) in terms of the integral structure on \(V = H^1(S, \mathbb{Z}) \otimes \mathbb{C}\) and …