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Nachr. 287(14–15), 1618–1623 (2014)\nBaltazar, H., Batista, R., Bezerra, K.: On the volume functional of compact manifolds with boundary with harmonic Weyl tensor, arxiv:1710.06247 (2017)\nBesse, A.L.: Einstein manifolds. Ergebnisse der Mathematik, 3 Folge, Band 10, Springer-Verlag, (1987)\nBernstein, J., Mettler, T.: Two-dimensional gradient Ricci solitons revisited. Int. Math. Res. Notices 1, 78–98 (2015)\nBrendle, S.: Rotational symmetry of self-similar solutions to the Ricci flow. Invent. Math. 194(3), 731–764 (2013)\nCao, H.D.: Recent progress on Ricci solitons, Recent advances in geometric analysis, 138, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, (2010)\nCao, H.D., Catino, G., Chen, Q., Mantegazza, C., Mazzieri, L.: Bach-flat gradient steady Ricci solitons. Calc. Var. Partial Differ. Equ. 49(1–2), 125–138 (2014)\nCao, H.D., Chen, Q.: On locally conformally flat gradient steady Ricci solitons. Trans. Amer. Math. 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Math. 242(1), 189–200 (2009)",{"EN":474},"In this paper, we study a three-dimensional Ricci-degenerate Riemannian manifold \n                \n                  \n                \n                $$(M^3,g)$$\n                \n               that admits a smooth nontrivial solution f to the equation \n                1\n                \n                  \n                \n                $$\\begin{aligned} \\nabla df=\\psi Rc+\\phi g, \\end{aligned}$$\n                \n              where \n                \n                  \n                \n                $$\\psi ,\\phi $$\n                \n               are given smooth functions of f, Rc is the Ricci tensor of g. Spaces of this type include various interesting classes, namely gradient Ricci solitons, m-quasi Einstein metrics, (vacuum) static spaces, V-static spaces, and critical point metrics. The m-quasi Einstein metrics and vacuum static spaces were previously studied in Jordan (Gen Relativ Gravit 41(9):2191–2280, 2009) and Kim and Shin (Math Nachr 292(8): 1727–1750, 2019), respectively. In this paper, we refine them and develop a general approach for the solutions of (1). We specify the shape of the metric g satisfying (1) when \n                \n                  \n                \n                $$\\nabla f$$\n                \n               is not a Ricci-eigen vector. Then we focus on the remaining three classes, namely gradient Ricci solitons, V-static spaces, and critical point metrics. 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When K\u002FF is a cyclic extension of degree \n                \n                  \n                \n                $$p^n$$\n                \n              , we determine the \n                \n                  \n                \n                $$\\mathbb {Z}\u002Fp^m\\mathbb {Z}[\\text {Gal}(K\u002FF)]$$\n                \n              -module structure of \n                \n                  \n                \n                $$K^\\times \u002FK^{\\times p^m}$$\n                \n              . With at most one exception, each indecomposable summand is cyclic and free over some quotient group of \n                \n                  \n                \n                $$\\text {Gal}(K\u002FF)$$\n                \n              . For fixed values of m and n, there are only finitely many possible isomorphism classes for the non-free indecomposable summand. These Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation provides insight into the structure of absolute Galois groups. More immediately, however, these results show that Galois cohomology is a context in which seemingly difficult module decompositions can practically be achieved: when \n                \n                  \n                \n                $$m,n>1$$\n                \n               the modular representation theory allows for an infinite number of indecomposable summands (with no known classification of indecomposable types), and yet the main result of this paper provides a complete decomposition over an infinite family of modules.",{"EN":720},"Galois module structure of the units modulo $$p^m$$ of cyclic extensions of degree $$p^n$$",{"VOID":722},"10.1007\u002Fs00229-022-01385-z","2025-01-24T23:57:36.297+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00229-022-01385-z",[726,747,762],{"id":727,"sortIndex":111,"researcher":20,"roles":728,"affiliations":729,"properties":744},"8e96fe65-d167-4793-a629-72b3cb26d27c",[276],[730],{"id":731,"sortIndex":21,"affiliation":732,"properties":741},"1753e65f-1884-4c9e-a300-a640857f8d66",{"id":733,"createTime":734,"updateTime":735,"relativeEntities":736,"slug":737,"properties":738,"entityType":41,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21},"1b1ef8c4-904e-4476-805e-93f53bc4908b","2024-04-22T05:46:33.951+00:00","2024-09-22T05:43:29.095+00:00",[],"Department-of-Mathematics-Wellesley-College-Wellesley-U-S-A",{"title":739},{"EN":740},"Department of Mathematics, Wellesley College, Wellesley, U.S.A",{"title":742},{"EN":743},"Department of Mathematics, Wellesley College, 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A riemannian flow\n                  \n                    \n                  \n                  \n$$\\mathfrak{F}$$\n\n                 on M is an oriented one dimensional foliation which admits a bundle-like metric. We give a caracterization of isometric flows as riemannian flows whose basic cohomology H\n                  b\n                  n−1\n                (M,\n                  \n                    \n                  \n                  \n$$\\mathfrak{F}$$\n\n                ) is non trivial in degree (n−1). A second caracterization involves the triviality of the central sheaf. 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