三维Tingley问题中的仿射锚与圆柱障碍
Affine Anchors and Cylinder Obstructions in the Three-Dimensional Tingley Problem
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中文总结 AI 辅助
本文研究三维实Banach空间中满射等距的仿射传播，给出Mazur--Ulam性质的源几何判据与线段饱和判据，并揭示圆柱情形下最终障碍为范数校准的逐点升级。
中文摘要 AI 辅助
设$X$为三维实Banach空间，$f:S_X\to S_Y$为满射等距。我们研究$f$在$S_X$的相对开子集上仿射公式的传播性质。有限族对径距离坐标可传播线性锚，除非出现三种明确描述的非退化情形：一个面、一个开面星或一族具有公共圆柱核的弦锥。我们证明，若范数不含固定方向的圆柱开锥，则每个开仿射锚都是线性的且是全局的。这给出了Mazur--Ulam性质的源几何判据，并涵盖了包括欧几里得双锥在内的其他非严格凸例子。我们还给出一个线段饱和判据，该判据证明了对于任意实Banach平面$Z$，每个棱柱$Z\oplus_\infty\mathbb R$均具有该性质。最后，在剩余的圆柱情形中，我们证明弦饱和同核网络不能局限于单个真射影商弧。随后我们表明，这种多弧困难与产生初始仿射锚的横向直纹部分达到相同的最终障碍：将环境开锥上的范数校准连同若干球面线段上的校准，升级为二维开补片上球面映射的逐点校准。
英文摘要
Let $X$ be a three-dimensional real Banach space and let $f:S_X\to S_Y$ be a surjective isometry. We study the propagation of an affine formula for $f$ on a relatively open part of $S_X$. A finite family of antipodal distance coordinates propagates a linear anchor except at three explicitly described degeneracies: a facet, an open face-star, or a family of chord cones with a common cylindrical kernel. We prove that, if the norm has no fixed-direction cylindrical open cone, every affine open anchor is linear and global. This yields a source-geometric criterion for the Mazur--Ulam property and covers, among other non-strictly-convex examples, the Euclidean double cone. We also give a segment-saturation criterion which proves the property for every prism $Z\oplus_\infty\mathbb R$, where $Z$ is an arbitrary real Banach plane. Finally, in the remaining cylindrical regime, we prove that a chord-saturated same-kernel network cannot be confined to one proper projective quotient arc. We then show that this multi-arc difficulty and the transverse-ruled part of producing an initial affine anchor reach the same final obstruction: upgrading norm calibration on an ambient open cone, together with calibration on a few spherical segments, to pointwise calibration of the sphere map on a two-dimensional open patch.
