{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:40:23Z","timestamp":1759970423949,"version":"build-2065373602"},"reference-count":6,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,17]],"date-time":"2025-01-17T00:00:00Z","timestamp":1737072000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSF of China","award":["11901100"],"award-info":[{"award-number":["11901100"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A problem that geometers have always been concerned with is when a closed manifold is isometric to a round sphere. A classical result shows that a closed locally conformally flat Einstein manifold is always isometric to a quotient of a round sphere. In this note, we provide the definitions of \u03c3k-curvatures and \u03c3k-Einstein manifolds, and we show that a closed \u03c3k-Einstein manifold under certain pinching conditions of a Weyl curvature and Einstein curvature is isometric to a quotient of a round sphere.<\/jats:p>","DOI":"10.3390\/axioms14010068","type":"journal-article","created":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T12:32:52Z","timestamp":1737376372000},"page":"68","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sphere Theorems for \u03c3k-Einstein Manifolds"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-1298-9433","authenticated-orcid":false,"given":"Jingyang","family":"Zhong","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinran","family":"Mu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Besse, A. (1987). Einstein Manifolds, Springer.","DOI":"10.1007\/978-3-540-74311-8"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/0926-2245(92)90014-E","article-title":"Positive Einstein metrics with small Ln\/2-norm of the Weyl tensor","volume":"2","author":"Singer","year":"1992","journal-title":"Differ. Geom. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.difgeo.2019.01.004","article-title":"A sphere theorem for Bach-flat manifolds with positive constant scalar curvature","volume":"64","author":"Fang","year":"2019","journal-title":"Differ. Geom. Its Appl."},{"key":"ref_4","first-page":"373","article-title":"On the Hessian of a function and the curvatures of its graph","volume":"20","author":"Reilly","year":"1973","journal-title":"Michigan Math. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1512\/iumj.1977.26.26036","article-title":"Applications of the Hessian operator in a Riemannian manifold","volume":"26","author":"Reilly","year":"1977","journal-title":"Indiana Univ. Math. J."},{"key":"ref_6","unstructured":"Yuan, W. (2015). The Geometry of Vacuum Static Spaces and Deformations of Scalar Curvature. [Ph.D. Thesis, University of California]."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/68\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:31:00Z","timestamp":1759919460000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/68"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,17]]},"references-count":6,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010068"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010068","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,1,17]]}}}