{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T00:02:34Z","timestamp":1769472154348,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,27]],"date-time":"2025-01-27T00:00:00Z","timestamp":1737936000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU-DDRSP2503"],"award-info":[{"award-number":["IMSIU-DDRSP2503"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The object of this paper is to find a vector field \u03be and a constant \u03bb on an n-dimensional compact Riemannian manifold Mn,g such that we obtain the Ricci soliton Mn,g,\u03be,\u03bb. In order to achieve this objective, we choose an isometric embedding provided in the work of Kuiper and Nash in the Euclidean space Rm,g\u00af and choose \u03be as the tangential component of a constant unit vector on Rm and call it a Kuiper\u2013Nash vector. If \u03c4 is the scalar curvature of the compact Riemannian manifold Mn,g with a Kuiper\u2013Nash vector \u03be, we show that if the integral of the function \u03be\u03c4 has a suitable lower bound containing a constant \u03bb, then Mn,g,\u03be,\u03bb is a Ricci soliton; we call this a Kuiper\u2013Nash Ricci soliton. We find a necessary and sufficient condition involving the scalar curvature \u03c4 under which a compact Kuiper\u2013Nash Ricci soliton Mn,g,\u03be,\u03bb is a trivial soliton. Finally, we find a characterization of an n-dimensional compact trivial Kuiper\u2013Nash Ricci soliton Mn,g,\u03be,\u03bb using an upper bound on the integral of div\u03be2 containing the scalar curvature \u03c4.<\/jats:p>","DOI":"10.3390\/axioms14020095","type":"journal-article","created":{"date-parts":[[2025,1,28]],"date-time":"2025-01-28T08:57:48Z","timestamp":1738054668000},"page":"95","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Extrinsic Geometry of a Riemannian Manifold and Ricci Solitons"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5901-2511","authenticated-orcid":false,"given":"Ibrahim","family":"Al-Dayel","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3700-8164","authenticated-orcid":false,"given":"Sharief","family":"Deshmukh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Carmo, M.P.D. (1992). Riemannian Geometry, Birkhauser.","DOI":"10.1007\/978-1-4757-2201-7"},{"key":"ref_2","unstructured":"Kobayashi, S., and Nomizu, K. (1963). Foundations of Differential Geometry, Interscience Publishers."},{"key":"ref_3","unstructured":"Kobayashi, S., and Nomizu, K. (1969). Interscience Tracts in Pure and Applied Mathematics. Foundations of Differential Geometry, Vol II, John Wiley & Sons."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/S1385-7258(59)50002-5","article-title":"Isometric and short imbeddings","volume":"21","author":"Kuiper","year":"1959","journal-title":"Indag. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s11401-005-0379-2","article-title":"Geometry of Ricci solitons","volume":"27","author":"Cao","year":"2006","journal-title":"Chin. Ann. Math. Ser. B"},{"key":"ref_6","first-page":"1","article-title":"Recent progress on Ricci solitons","volume":"11","author":"Cao","year":"2010","journal-title":"Adv. Lect. Math. (ALM)"},{"key":"ref_7","first-page":"35","article-title":"Global Nash-Kuiper theorem for compact manifolds","volume":"122","author":"Cao","year":"2022","journal-title":"J. Diff. Geom."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"93","DOI":"10.5937\/KgJMath1701093C","article-title":"Rectifying submanifolds of Riemannian manifolds and torqued vector fields","volume":"41","author":"Chen","year":"2017","journal-title":"Kragujev. J. Math."},{"key":"ref_9","unstructured":"Chow, B., Lu, P., and Ni, L. (2010). Graduate Studies in Mathematics. Hamilton\u2019s Ricci Flow, AMS Scientific Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1090\/S0002-9904-1974-13457-9","article-title":"Manifolds of Riemannian metrics with prescribed scalar curvature","volume":"80","author":"Fischer","year":"1974","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1090\/bull\/1551","article-title":"Geometric, Algebraic, and Analytic Descendants of Nash Isometric embedding theorems","volume":"54","author":"Gromov","year":"2017","journal-title":"Bull. Am. Math. Soc. (New Ser.)"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"20","DOI":"10.2307\/1969989","article-title":"The imbedding problem for Riemannian manifolds","volume":"63","author":"Nash","year":"1956","journal-title":"Ann. Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Chen, B.-Y. (1983). Total Mean Curvature and Submanifolds of Finite Type, World Scientific.","DOI":"10.1142\/0065"},{"key":"ref_14","unstructured":"Chen, B.-Y. (1973). Geometry of Submanifolds, Marcel Dekker."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/PL00012538","article-title":"Geometry of position functions of Riemannian submanifolds in pseudo-Euclidean space","volume":"74","author":"Chen","year":"2002","journal-title":"J. Geom."},{"key":"ref_16","unstructured":"Chen, B.-Y. (1981). Geometry of Submanifolds and Its Applications, Science University of Tokyo."},{"key":"ref_17","first-page":"9","article-title":"More on convolution of Riemannian manifolds","volume":"44","author":"Chen","year":"2003","journal-title":"Beitr\u00e4ge Algebra Geom."},{"key":"ref_18","first-page":"281","article-title":"Constant-ratio space-like submanifolds in pseudo-Euclidean space","volume":"29","author":"Chen","year":"2003","journal-title":"Houston J. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.36890\/iejg.584566","article-title":"Differential geometry of rectifying submanifolds","volume":"9","author":"Chen","year":"2016","journal-title":"Int. Electron. J. Geom."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4","DOI":"10.36890\/iejg.1216024","article-title":"Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map","volume":"16","author":"Chen","year":"2023","journal-title":"Int. Elect. J. Geom."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"306","DOI":"10.2307\/2372684","article-title":"On the total curvature of immersed manifolds","volume":"79","author":"Chern","year":"1957","journal-title":"Am. J. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1307\/mmj\/1028998005","article-title":"On the total curvature of immersed manifolds II","volume":"5","author":"Chern","year":"1958","journal-title":"Mich. Math. J."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Rovenskii, V. (1998). Foliations on Riemannian manifolds and Submanifolds, Birkhauser.","DOI":"10.1007\/978-1-4612-4270-3_1"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/S0252-9602(17)30560-X","article-title":"Submanifolds of product Riemannian manifold","volume":"20","author":"Xu","year":"2000","journal-title":"Acta Math. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2413","DOI":"10.1090\/S0002-9947-96-01632-7","article-title":"Total absolute curvature and Tightness of noncompact manifolds","volume":"348","author":"Gemmeren","year":"1996","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF01871945","article-title":"On total absolute curvature of nonclosed submanifolds","volume":"2","author":"Wintgen","year":"1984","journal-title":"Ann. Glob. Anal. Geom."},{"key":"ref_27","first-page":"109","article-title":"Nonlinear analysis in geometry","volume":"33","author":"Yau","year":"1987","journal-title":"Enseign. Math."},{"key":"ref_28","unstructured":"Yau, S.-T. (1992). Open problems in geometry. Chern\u2014A Great Geometer of the Twentieth Century, International Press."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1215\/S0012-7094-39-00507-7","article-title":"Isometric embedding of flat manifolds in Euclidean space","volume":"5","author":"Tompkins","year":"1939","journal-title":"Duke Math. J."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1017\/S0004972700011801","article-title":"Isometric immersion of a compact Riemannian manifold into a Euclidean space","volume":"46","author":"Deshmukh","year":"1992","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1090\/S0002-9939-1973-0375173-3","article-title":"Isometric embedding of a compact Riemannian manifold into Euclidean space","volume":"40","author":"Jacabowitz","year":"1973","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_32","unstructured":"Yano, K. (1970). Integral Formulas in Riemannian Geometry, Marcel Dekker Inc."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/95\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:37:07Z","timestamp":1759919827000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/95"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,27]]},"references-count":32,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["axioms14020095"],"URL":"https:\/\/doi.org\/10.3390\/axioms14020095","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,27]]}}}