differential geometry Archives

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 …

Demystification of the Willmore integrand

The Willmore energy for a surface $S$ in Euclidean 3-space is defined as $\tilde{W}(S) = \int_S \mu^2\omega_S$ , where $\mu$ is the mean curvature of $S$ and $\omega_S$ its area form. It’s known to be invariant under the conformal transformations (whereas the mean curvature itself is not). White, and later Bryant noticed that the 2-form $\Omega_S = (\mu^2-K)\omega_S$ , where $K$ stands …

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 …