@article{Gigli2019,
  author        = {Gigli, Nicola},
  title         = {{Riemann} curvature tensor on {RCD} spaces and possible applications},
  journal       = {Comptes Rendus Math{\'e}matique},
  volume        = {357},
  number        = {7},
  pages         = {613--619},
  year          = {2019},
  doi           = {10.1016/j.crma.2019.06.003},
  eprint        = {1902.02282},
  archivePrefix = {arXiv},
  primaryClass  = {math.DG},
  url           = {https://arxiv.org/abs/1902.02282v1},
  note          = {Numbering follows arXiv:1902.02282v1}
}

@article{AmbrosioTrevisan2014,
  author        = {Ambrosio, Luigi and Trevisan, Dario},
  title         = {Well-posedness of {Lagrangian} flows and continuity equations in metric measure spaces},
  journal       = {Analysis \& PDE},
  volume        = {7},
  number        = {5},
  pages         = {1179--1234},
  year          = {2014},
  doi           = {10.2140/apde.2014.7.1179},
  eprint        = {1402.4788},
  archivePrefix = {arXiv},
  primaryClass  = {math.FA},
  url           = {https://arxiv.org/abs/1402.4788v3}
}

@article{GigliRajalaSturm2016,
  author        = {Gigli, Nicola and Rajala, Tapio and Sturm, Karl-Theodor},
  title         = {Optimal maps and exponentiation on finite-dimensional spaces with {Ricci} curvature bounded from below},
  journal       = {The Journal of Geometric Analysis},
  volume        = {26},
  number        = {4},
  pages         = {2914--2929},
  year          = {2016},
  doi           = {10.1007/s12220-015-9654-y},
  eprint        = {1305.4849},
  archivePrefix = {arXiv},
  primaryClass  = {math.DG},
  url           = {https://arxiv.org/abs/1305.4849v1}
}

@article{Rajala2012,
  author        = {Rajala, Tapio},
  title         = {Improved geodesics for the reduced curvature-dimension condition in branching metric spaces},
  journal       = {Discrete and Continuous Dynamical Systems},
  volume        = {33},
  number        = {7},
  pages         = {3043--3056},
  year          = {2013},
  doi           = {10.3934/dcds.2013.33.3043},
  eprint        = {1202.6514},
  archivePrefix = {arXiv},
  primaryClass  = {math.DG},
  url           = {https://arxiv.org/abs/1202.6514}
}

@article{GigliTamanini2021,
  author        = {Gigli, Nicola and Tamanini, Luca},
  title         = {Second order differentiation formula on {$\mathrm{RCD}^{*}(K,N)$} spaces},
  journal       = {Journal of the European Mathematical Society},
  volume        = {23},
  number        = {5},
  pages         = {1727--1795},
  year          = {2021},
  doi           = {10.4171/JEMS/1042},
  eprint        = {1802.02463},
  archivePrefix = {arXiv},
  primaryClass  = {math.AP},
  url           = {https://arxiv.org/abs/1802.02463v1},
  note          = {Numbering follows the 2018 preprint, arXiv:1802.02463v1}
}

@article{DPG2018,
  author        = {De Philippis, Guido and Gigli, Nicola},
  title         = {Non-collapsed spaces with {Ricci} curvature bounded from below},
  journal       = {Journal de l'{\'E}cole polytechnique --- Math{\'e}matiques},
  volume        = {5},
  pages         = {613--650},
  year          = {2018},
  doi           = {10.5802/jep.80},
  eprint        = {1708.02060},
  archivePrefix = {arXiv},
  primaryClass  = {math.MG},
  url           = {https://arxiv.org/abs/1708.02060v3}
}

@article{Deng2020,
  author        = {Deng, Qin},
  title         = {{H{\"o}lder} continuity of tangent cones in {$\mathrm{RCD}(K,N)$} spaces and applications to non-branching},
  journal       = {Geometry \& Topology},
  volume        = {29},
  pages         = {1037--1114},
  year          = {2025},
  doi           = {10.2140/gt.2025.29.1037},
  eprint        = {2009.07956},
  archivePrefix = {arXiv},
  primaryClass  = {math.DG},
  url           = {https://arxiv.org/abs/2009.07956v2}
}

@article{BPS2021,
  author        = {Bru{\`e}, Elia and Pasqualetto, Enrico and Semola, Daniele},
  title         = {Rectifiability of {$\mathrm{RCD}(K,N)$} spaces via {$\delta$}-splitting maps},
  journal       = {Annales Fennici Mathematici},
  volume        = {46},
  number        = {1},
  pages         = {465--482},
  year          = {2021},
  doi           = {10.5186/aasfm.2021.4627},
  eprint        = {2001.07911},
  archivePrefix = {arXiv},
  primaryClass  = {math.MG},
  url           = {https://arxiv.org/abs/2001.07911v1}
}

@article{Savare2014,
  author        = {Savar{\'e}, Giuseppe},
  title         = {Self-improvement of the {Bakry--{\'E}mery} condition and {Wasserstein} contraction of the heat flow in {$\mathrm{RCD}(K,\infty)$} metric measure spaces},
  journal       = {Discrete and Continuous Dynamical Systems},
  volume        = {34},
  number        = {4},
  pages         = {1641--1661},
  year          = {2014},
  doi           = {10.3934/dcds.2014.34.1641},
  eprint        = {1304.0643},
  archivePrefix = {arXiv},
  primaryClass  = {math.FA},
  url           = {https://arxiv.org/abs/1304.0643v2}
}

@article{BrenaGigli2024,
  author        = {Brena, Camillo and Gigli, Nicola},
  title         = {Fine representation of {Hessian} of convex functions and {Ricci} tensor on {RCD} spaces},
  journal       = {Potential Analysis},
  volume        = {62},
  pages         = {703--737},
  year          = {2025},
  doi           = {10.1007/s11118-024-10153-5},
  eprint        = {2310.07536},
  archivePrefix = {arXiv},
  primaryClass  = {math.MG},
  url           = {https://arxiv.org/abs/2310.07536v1},
  note          = {Remark numbering follows arXiv:2310.07536v1}
}

@article{BB2018,
  author        = {Bj{\"o}rn, Anders and Bj{\"o}rn, Jana},
  title         = {{Poincar{\'e}} inequalities and {Newtonian Sobolev} functions on noncomplete metric spaces},
  journal       = {Journal of Differential Equations},
  volume        = {266},
  pages         = {44--69},
  year          = {2019},
  doi           = {10.1016/j.jde.2018.07.029},
  eprint        = {1705.02253},
  archivePrefix = {arXiv},
  primaryClass  = {math.AP},
  url           = {https://arxiv.org/abs/1705.02253v2},
  note          = {The linked arXiv version includes a correction appended in 2020}
}

@article{CGP2022,
  author        = {Caputo, Emanuele and Gigli, Nicola and Pasqualetto, Enrico},
  title         = {Parallel transport on non-collapsed {$\mathrm{RCD}(K,N)$} spaces},
  journal       = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume        = {819},
  pages         = {135--204},
  year          = {2025},
  doi           = {10.1515/crelle-2024-0082},
  eprint        = {2108.07531},
  archivePrefix = {arXiv},
  primaryClass  = {math.DG},
  url           = {https://arxiv.org/abs/2108.07531}
}

@article{Gigli2018,
  author        = {Gigli, Nicola},
  title         = {Nonsmooth differential geometry---an approach tailored for spaces with {Ricci} curvature bounded from below},
  journal       = {Memoirs of the American Mathematical Society},
  volume        = {251},
  number        = {1196},
  pages         = {v+161},
  year          = {2018},
  doi           = {10.1090/memo/1196},
  url           = {https://bookstore.ams.org/memo-251-1196}
}

@article{Petrunin2011,
  author        = {Petrunin, Anton},
  title         = {{Alexandrov} meets {Lott--Villani--Sturm}},
  journal       = {M{\"u}nster Journal of Mathematics},
  volume        = {4},
  pages         = {53--64},
  year          = {2011},
  eprint        = {1003.5948},
  archivePrefix = {arXiv},
  url           = {https://www.uni-muenster.de/FB10/mjm/vol_4/mjm_vol_4_03.pdf}
}

@article{ZhangZhu2010,
  author        = {Zhang, Hui-Chun and Zhu, Xi-Ping},
  title         = {{Ricci} curvature on {Alexandrov} spaces and rigidity theorems},
  journal       = {Communications in Analysis and Geometry},
  volume        = {18},
  number        = {3},
  pages         = {503--553},
  year          = {2010},
  eprint        = {0912.3190},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/0912.3190}
}

@article{KMS2001,
  author        = {Kuwae, Kazuhiro and Machigashira, Yoshiroh and Shioya, Takashi},
  title         = {{Sobolev} spaces, {Laplacian}, and heat kernel on {Alexandrov} spaces},
  journal       = {Mathematische Zeitschrift},
  volume        = {238},
  number        = {2},
  pages         = {269--316},
  year          = {2001},
  doi           = {10.1007/s002090100252},
  url           = {https://link.springer.com/article/10.1007/s002090100252}
}

@incollection{Petrunin2007,
  author        = {Petrunin, Anton},
  title         = {Semiconcave functions in {Alexandrov}'s geometry},
  booktitle     = {Surveys in Differential Geometry. Vol. XI},
  publisher     = {International Press},
  address       = {Somerville, MA},
  pages         = {137--201},
  year          = {2007},
  doi           = {10.4310/SDG.2006.v11.n1.a6},
  eprint        = {1304.0292},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/1304.0292},
  note          = {Updated arXiv version submitted in 2013 and revised in 2015}
}

@book{AKP2024,
  author        = {Alexander, Stephanie and Kapovitch, Vitali and Petrunin, Anton},
  title         = {{Alexandrov} Geometry: Foundations},
  series        = {Graduate Studies in Mathematics},
  volume        = {236},
  publisher     = {American Mathematical Society},
  year          = {2024},
  doi           = {10.1090/gsm/236},
  isbn          = {978-1-4704-7302-0},
  url           = {https://anton-petrunin.github.io/book/arXiv.pdf},
  note          = {Numbering follows the updated online version accessed 24 September 2026}
}

@article{EKOV2026,
  author        = {Er{\"o}s, Darius and Kunzinger, Michael and Ohanyan, Argam and Vardabasso, Alessio},
  title         = {Distributional sectional curvature bounds for {Riemannian} metrics of low regularity},
  journal       = {The Journal of Geometric Analysis},
  volume        = {36},
  eid           = {92},
  year          = {2026},
  doi           = {10.1007/s12220-026-02339-7},
  url           = {https://link.springer.com/article/10.1007/s12220-026-02339-7},
  note          = {Article 92}
}

@article{ErbarKuwadaSturm2015,
  author        = {Erbar, Matthias and Kuwada, Kazumasa and Sturm, Karl-Theodor},
  title         = {On the equivalence of the entropic curvature-dimension condition and {Bochner}'s inequality on metric measure spaces},
  journal       = {Inventiones Mathematicae},
  volume        = {201},
  pages         = {993--1071},
  year          = {2015},
  doi           = {10.1007/s00222-014-0563-7},
  eprint        = {1303.4382},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/1303.4382v2}
}

@article{JiangLiZhang2016,
  author        = {Jiang, Renjin and Li, Huaiqian and Zhang, Huichun},
  title         = {Heat kernel bounds on metric measure spaces and some applications},
  journal       = {Potential Analysis},
  volume        = {44},
  pages         = {601--627},
  year          = {2016},
  doi           = {10.1007/s11118-015-9521-2},
  eprint        = {1407.5289},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/1407.5289v2}
}

@article{GigliDifferentialStructure,
  author        = {Gigli, Nicola},
  title         = {Lecture notes on differential calculus on {RCD} spaces},
  journal       = {Publications of the Research Institute for Mathematical Sciences},
  volume        = {54},
  number        = {4},
  pages         = {855--918},
  year          = {2018},
  doi           = {10.4171/PRIMS/54-4-4},
  eprint        = {1703.06829},
  archivePrefix = {arXiv},
  url           = {https://ems.press/content/serial-article-files/41329}
}

@article{GigliHan2015,
  author        = {Gigli, Nicola and Han, Bang-Xian},
  title         = {The continuity equation on metric measure spaces},
  journal       = {Calculus of Variations and Partial Differential Equations},
  volume        = {53},
  number        = {1--2},
  pages         = {149--177},
  year          = {2015},
  doi           = {10.1007/s00526-014-0744-7},
  eprint        = {1406.6350},
  archivePrefix = {arXiv},
  primaryClass  = {math.AP},
  url           = {https://arxiv.org/abs/1406.6350v1}
}

@article{KapovitchKettererSturm,
  author        = {Kapovitch, Vitali and Ketterer, Christian and Sturm, Karl-Theodor},
  title         = {On gluing {Alexandrov} spaces with lower {Ricci} curvature bounds},
  journal       = {Communications in Analysis and Geometry},
  volume        = {31},
  number        = {6},
  pages         = {1529--1564},
  year          = {2023},
  doi           = {10.4310/CAG.2023.v31.n6.a6},
  eprint        = {2003.06242},
  archivePrefix = {arXiv},
  primaryClass  = {math.DG},
  url           = {https://arxiv.org/abs/2003.06242v1},
  note          = {Theorem numbering follows the 2020 preprint, arXiv:2003.06242v1}
}

@misc{WeakHessianCompanion,
  author = {{OpenAI}},
  title = {{Weak Hessian bounds along every geodesic in RCD spaces}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf}{OAI:Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026}},
  year = {2026},
  note = {Theorem 1.1}
}

@article{LebedevaPetrunin2024,
 author = {Lebedeva, Nina and Petrunin, Anton},
 title = {Curvature tensor of smoothable {Alexandrov} spaces},
 journal = {Geometry \& Topology}, volume = {28}, pages = {3869--3907},
 year = {2024}, doi = {10.2140/gt.2024.28.3869},
 url = {https://arxiv.org/abs/2202.13420v3}
}

@incollection{Leonard2017,
 author = {L{\'e}onard, Christian},
 title = {On the convexity of the entropy along entropic interpolations},
 booktitle = {Measure Theory in Non-Smooth Spaces},
 editor = {Gigli, Nicola}, publisher = {De Gruyter Open},
 pages = {194--242}, year = {2017},
 doi = {10.1515/9783110550832-006},
 url = {https://arxiv.org/abs/1310.1274v1}
}

@article{AGSWeakGradients2013,
 author = {Ambrosio, Luigi and Gigli, Nicola and Savar{\'e}, Giuseppe},
 title = {Density of {Lipschitz} functions and equivalence of weak gradients in metric measure spaces},
 journal = {Revista Matem{\'a}tica Iberoamericana},
 volume = {29}, number = {3}, pages = {969--996}, year = {2013},
 doi = {10.4171/rmi/746},
 url = {https://ems.press/content/serial-article-files/38442}
}

@article{LottVillani2009,
  author = {Lott, John and Villani, C{\'e}dric},
  title = {{Ricci} curvature for metric-measure spaces via optimal transport},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {169},
  number = {3},
  pages = {903--991},
  year = {2009},
  doi = {10.4007/annals.2009.169.903},
  url = {https://annals.math.princeton.edu/2009/169-3/p04}
}

@article{Sturm2006I,
  author = {Sturm, Karl-Theodor},
  title = {On the geometry of metric measure spaces. {I}},
  journal = {Acta Mathematica},
  volume = {196},
  number = {1},
  pages = {65--131},
  year = {2006},
  doi = {10.1007/s11511-006-0002-8},
  url = {https://www.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Karl-Theodor_Sturm/papers/paper47-new.pdf}
}

@article{Sturm2006II,
  author = {Sturm, Karl-Theodor},
  title = {On the geometry of metric measure spaces. {II}},
  journal = {Acta Mathematica},
  volume = {196},
  number = {1},
  pages = {133--177},
  year = {2006},
  doi = {10.1007/s11511-006-0003-7},
  url = {https://www.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Karl-Theodor_Sturm/papers/paper50-new.pdf}
}

@article{AGS2014,
  author = {Ambrosio, Luigi and Gigli, Nicola and Savar{\'e}, Giuseppe},
  title = {Metric measure spaces with {Riemannian Ricci} curvature bounded from below},
  journal = {Duke Mathematical Journal},
  volume = {163},
  number = {7},
  pages = {1405--1490},
  year = {2014},
  doi = {10.1215/00127094-2681605},
  eprint = {1109.0222},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.1215/00127094-2681605}
}

@article{AGMR2015,
  author = {Ambrosio, Luigi and Gigli, Nicola and Mondino, Andrea and Rajala, Tapio},
  title = {{Riemannian Ricci} curvature lower bounds in metric measure spaces with {$\sigma$}-finite measure},
  journal = {Transactions of the American Mathematical Society},
  volume = {367},
  number = {7},
  pages = {4661--4701},
  year = {2015},
  doi = {10.1090/S0002-9947-2015-06111-X},
  eprint = {1207.4924},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1207.4924v2}
}

@article{KettererMondino2018,
  author = {Ketterer, Christian and Mondino, Andrea},
  title = {Sectional and intermediate {Ricci} curvature lower bounds via optimal transport},
  journal = {Advances in Mathematics},
  volume = {329},
  pages = {781--818},
  year = {2018},
  doi = {10.1016/j.aim.2018.01.024},
  eprint = {1610.03339},
  archivePrefix = {arXiv},
  url = {https://cvgmt.sns.it/media/doc/paper/3198/intermediate_26022018.pdf}
}