{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:03:08Z","timestamp":1760241788516,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2018,9,11]],"date-time":"2018-09-11T00:00:00Z","timestamp":1536624000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007446","name":"King Khalid University","doi-asserted-by":"publisher","award":["N.A."],"award-info":[{"award-number":["N.A."]}],"id":[{"id":"10.13039\/501100007446","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we obtain the upper bounds for the normalized    \u03b4   -Casorati curvatures and generalized normalized    \u03b4   -Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the necessary and sufficient condition for a Sasaki-like statistical manifold to be    \u03b7   -Einstein. Finally, we provide the condition under which the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations.<\/jats:p>","DOI":"10.3390\/e20090690","type":"journal-article","created":{"date-parts":[[2018,9,11]],"date-time":"2018-09-11T11:40:02Z","timestamp":1536666002000},"page":"690","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Pinching Theorems for Statistical Submanifolds in Sasaki-Like Statistical Space Forms"],"prefix":"10.3390","volume":"20","author":[{"given":"Ali H.","family":"Alkhaldi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 62529, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohd.","family":"Aquib","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi 110025, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3895-7548","authenticated-orcid":false,"given":"Aliya Naaz","family":"Siddiqui","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi 110025, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammad Hasan","family":"Shahid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi 110025, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Amari, S. 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