A construction of curvature-adapted hypersurfaces in the product of symmetric spaces

TL;DR

The paper addresses constructing curvature-adapted hypersurfaces in the product $G_1/K_1\times G_2/K_2$ of symmetric spaces by pairing curvature-adapted hypersurfaces from each factor and deforming along a small curve on $S^1$; the main construction yields a tubular-like hypersurface $M_1\times M_2\times S^1$ whose immersion $f$ is guaranteed for sufficiently small deformations. Theorem A establishes that $f$ is an immersion and the resulting hypersurface is curvature-adapted, with explicit eigenvalues of the shape operator $A$ and the normal Jacobi operator $\widetilde{R}(\overline{\bm{N}})$, summarized in Table 1, and a formula for the product angle function $C$. The framework relies on the common eigenspace decomposition arising from commuting $A$ and $\widetilde{R}(\bm{N})$, and utilizes strongly $M$-Jacobi fields to derive the precise spectral data. This work extends tubes and curvature-adapted examples to a product setting, providing a large family of explicit curvature-adapted hypersurfaces with computable extrinsic data.

Abstract

In this paper, we give a construction of curvature-adapted hypersurfaces in the product $G_1/K_1\times G_2/K_2$ of (Riemannian) symmetric spaces $G_i/K_i$ ($i=1,2$). By this construction, we obtain many examples of curvature-adapted hypersurfaces in $G_1/K_1\times G_2/K_2$. Also, we calculate the eigenvalues of the shape operator and the normal Jacobi operator of the curvature-adapted hypersurfaces obtained by this construction.