On isometric embeddings of metric spaces

• Autores: Manuel Mellado Cuerno
• Directores de la Tesis: Luis Guijarro Santamaría (dir. tes.)
• Lectura: En la Universidad Autónoma de Madrid ( España ) en 2024
• Idioma: inglés
• Número de páginas: 90
• Títulos paralelos:
  • Sobre encajes isométricos de espacios métricos
• Tribunal Calificador de la Tesis: Gabino González Díez (presid.) , David González Álvaro (secret.) , Wilderich Tuschmann (voc.)
• Enlaces
  • Tesis en acceso abierto en: Biblos-e Archivo
• Resumen

  • Extracting and describing geometric and topological properties of certain metric spaces is often a challenging and costly task. Therefore, employing techniques that do not directly engage with these objects is often a wise strategy. One of the most commonly used approaches involves embedding these spaces into other spaces known as ambient spaces.

    The goal of this thesis is to work with isometric embeddings: the distance function of the ambient space restricted to the image of the embedding of our initial space must coincide with the distance function of the original metric space.

    There are numerous results concerning such embeddings, and some have gained considerable success due to their signi cance. A prime example is the embedding of smooth Riemannian manifolds by Nash [9]. He proved that every smooth Riemannian manifold can be isometrically embedded into a suciently high-dimensional Euclidean space. Another example, equally important and dating back over a century, is the result established by Hilbert [8], where he stated that there is no complete isometric copy of hyperbolic space within R3. Undoubtedly, one could provide an endless list of key results in various elds where such embeddings play a fundamental role, highlighting the importance of obtaining results related to them.

    In this dissertation, we will study two speci c isometric embeddings: the Kuratowski embedding and the canonical embedding into Wasserstein-type spaces.

    Regarding the rst one, we will show some upper and lower bounds for the Filling Radius (an invariant presented by Gromov in [7]) showing the universal positivity of the invariant and upper bounds for Riemannian submersions and submetries. Moreover, we will work with the Intermediate Filling Radius and prove their universal positivity as well as giving a bound for the value of the Intermediate Filling Radius of a manifold M regarding the one of N in the presence of a Lipschitz function f : M ! N between them.

    On the second part of the thesis, we will introduce the notion of the reach of a subset (in the sense of Federer [6]) but for the context of isometric embeddings. Firstly, for the Kuratowski embedding and secondly for the embedding into three Wassersteintype spaces: the canonical Wasserstein space, the Orlicz-Wassertein and the space of persistence diagrams