Rescaled Reflective Submanifolds of Symmetric Spaces

Definitions and mathematical statement

Let $\overline N$ be a semisimple Riemannian symmetric space and $N=\exp(\mathfrak p)$ the simply connected immersed reflective submanifold in the source. In the usual terminology, reflective submanifolds arise from fixed-point components of involutive ambient isometries, with the appropriate local/cover interpretation for the given immersion, and are totally geodesic. The basepoint, tangent subspace $\mathfrak p$ and global exponential convention are not specified.

Write $g_N$ for the induced metric and $N_R$ for the stated metric rescaling. The task is to decide local and global existence of an immersion \[ f:N_R\longrightarrow\overline N,\qquad f^*\overline g=g_R, \] subject to preservation of normal curvature and the $V_{\mathfrak p}$-submanifold condition. The parameter regimes are \[ R>1\quad\text{for compact type},\qquad 0<R<1\quad\text{for noncompact type}, \] where positivity is required for a Riemannian metric rescaling. The record prints $R<1$ in the latter case but does not spell out positivity or whether $g_R=R g_N$, $R^2g_N$, or another normalization. That convention is not guessed.

Local existence means an immersion on an appropriate neighborhood in $N_R$; global existence means one defined on the entire prescribed domain. A local solution is not a global one, and an immersion need not be injective, embedded or proper. No such strengthening is added. Preservation of normal curvature must compare the new immersion with the original reflective inclusion via the intended identification of normal bundles and their curvature tensors. It is not silently replaced by flatness of the normal bundle.

The symbol $V_{\mathfrak p}$ is undefined. It ordinarily signals an admissible tangent-plane/Gauss-map condition depending on $\mathfrak p$, but the relevant orbit, structure group and compatibility need to be recovered. Neither isometry alone nor a guessed full Grassmannian condition supplies it. The question therefore asks for existence/nonexistence by the intended $N,R$ and structural data, not a universal affirmative theorem for all symmetric spaces.

The literature triage checked 2026-08-21 reports obstructions to the compact analogue, but says a general noncompact global nonexistence theorem was explicitly not obtained and the simply connected hyperbolic case remains outstanding in the Hilbert-type program. It does not reproduce the exact compact obstruction hypotheses or the hyperbolic dimensions/codimensions. These are attributed progress limitations, not assertions that all local cases or every compact case have a uniform answer. The open label and editorial verification remain dated metadata.

Short English statement

Determine local and global isometric immersions of the stated rescaled reflective submanifolds, preserving the normal-curvature and $V_{\mathfrak p}$ conditions. Compact and noncompact parameter ranges are kept separate. The scaling convention and tangent-plane condition are undefined, and local existence is not treated as global completeness or an embedding.

Sources

• Local UnsolvedMath registry, permanent ID 30000763, source code catalogue entry, "Rescaled Reflective Submanifolds of Symmetric Spaces," including original and cleaned local/global questions and both parameter regimes.
• Progress in Surface Theory (2008), Oberwolfach Reports, pp. 1299--1376, DOI 10.4171/owr/2007/24: https://doi.org/10.4171/owr/2007/24. The citation uses 2008, while the DOI and proposed-year metadata use 2007.
• Supplied reference "Brander, Hilbert non-immersion generalizations": https://arxiv.org/abs/0710.0507.
• Literature triage checked 2026-08-21; record updated 2026-09-16. No external source has been visited or independently checked here.