Positive Weighted Curvatures on Compact Manifolds with Weighted Convex Boundary and Proper Manifolds

Abstract

Given a Riemannian manifold (M,g) equipped with a positive density function f = e^−φ, we study the weighted sectional curvature sec̅_φ(U,V) and the k-th intermediate weighted Ricci curvature Ric̅_k,φ(U,V). We show that sec̅_φ(U,V) and Ric̅_k,φ(U,V), together with the weighted second fundamental form I Ĩ, play central roles in controlling the topology of the manifold. Using the theory of C̃(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 sec̅_φ(U,V) and Ric̅_k,φ(U,V) provide effective tools in Riemannian geometry.

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BibTeX RIS

BibTeX

@misc{xu2026positive,
  title = {{Positive Weighted Curvatures on Compact Manifolds with Weighted Convex Boundary and Proper Manifolds}},
  author = {Ruifeng Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30527},
  eprint = {2609.30527},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30527},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Positive Weighted Curvatures on Compact Manifolds with Weighted Convex Boundary and Proper Manifolds
AU  - Ruifeng Xu
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30527
UR  - https://arxiv.org/abs/2609.30527
AB  - Given a Riemannian manifold (M,g) equipped with a positive density function f = e^−φ, we study the weighted sectional curvature sec̅_φ(U,V) and the k-th intermediate weighted Ricci curvature Ric̅_k,φ(U,V). We show that sec̅_φ(U,V) and Ric̅_k,φ(U,V), together with the weighted second fundamental form I Ĩ, play central roles in controlling the topology of the manifold. Using the theory of C̃(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 sec̅_φ(U,V) and Ric̅_k,φ(U,V) provide effective tools in Riemannian geometry.
ER  -

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