{
  "abstract": "Given a Riemannian manifold $(M,g)$ equipped with a positive density function $f = e^{-φ}$, we study the weighted sectional curvature $\\overline{\\sec}_φ(U,V)$ and the $k$-th intermediate weighted Ricci curvature $\\overline{\\text{Ric}}_{k,φ}(U,V)$. We show that $\\overline{\\sec}_φ(U,V)$ and $\\overline{\\text{Ric}}_{k,φ}(U,V)$, together with the weighted second fundamental form $\\widetilde{\\mathrm{I\\!I}}$, play central roles in controlling the topology of the manifold. Using the theory of $\\widetilde{C}(k)$ functions, variational methods, and Morse theory, we prove that by imposing suitable restrictions on these quantities, an $n$-dimensional compact manifold with boundary or a proper open $n$-dimensional manifold is diffeomorphic to an $n$-ball or to a certain CW complex. This shows that such rigidity phenomena is not confined in the unweighted case, but persist in the weighted setting as well. Lastly, we prove a weighted analogue of Frankel's Theorem under positive intermediate weighted Ricci curvature. These results extend classical theorems from the unweighted setting to the weighted case, demonstrating that $\\overline{\\sec}_φ(U,V)$ and $\\overline{\\text{Ric}}_{k,φ}(U,V)$ provide effective tools in Riemannian geometry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30527",
  "author_total": 1,
  "authors": [
    {
      "name": "Ruifeng Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6r8xtbrk8h4da.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30527",
  "fulltext_url": null,
  "id": "6r8xtbrk8h4da",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30527",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214654944",
  "pdf_url": "https://arxiv.org/pdf/2609.30527",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6r8xtbrk8h4da.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30527",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30527"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Positive Weighted Curvatures on Compact Manifolds with Weighted Convex Boundary and Proper Manifolds",
  "title_text": "Positive Weighted Curvatures on Compact Manifolds with Weighted Convex Boundary and Proper Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/6r8xtbrk8h4da",
  "venue": "arXiv",
  "versions": []
}