Submanifolds with constant scalar curvature in a space form

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• 作者：Jogli G. Araú ; jo ^{jogli@mat.ufcg.edu.br" class="auth_mail" title="E-mail the corresponding author} ; Henrique F. de Lima ; ^{henrique@mat.ufcg.edu.br" class="auth_mail" title="E-mail the corresponding author} ; Fá ; bio R. dos Santos ^{fabio@mat.ufcg.edu.br" class="auth_mail" title="E-mail the corresponding author} ; Marco Antonio L. Velá ; squez ^{marco.velasquez@mat.ufcg.edu.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Riemannian space forms ; Complete submanifolds ; Parallel normalized mean curvature vector field ; Constant normalized scalar curvature ; Clifford torus ; Circular and hyperbolic cylinders
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：447
• 期：1
• 页码：488-498
• 全文大小：313 K

文摘

We deal with complete submanifolds class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306291&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306291&_rdoc=1&_issn=0022247X&md5=24c0aadbae874e4aaa7973721c8db4df" title="Click to view the MathML source">Mⁿclass="mathContainer hidden">class="mathCode">$M^n$ having constant positive scalar curvature and immersed with parallel normalized mean curvature vector field in a Riemannian space form class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306291&_mathId=si104.gif&_user=111111111&_pii=S0022247X16306291&_rdoc=1&_issn=0022247X&md5=17f2021af5c738feb1e24d284b6d523b">class="imgLazyJSB inlineImage" height="17" width="39" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306291-si104.gif">class="mathContainer hidden">class="mathCode">${\mathbb{Q}}_c^{n+p}$ of constant sectional curvature class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306291&_mathId=si20.gif&_user=111111111&_pii=S0022247X16306291&_rdoc=1&_issn=0022247X&md5=d8f836e263482921d78edb75196251cd" title="Click to view the MathML source">c∈{1,0,−1}class="mathContainer hidden">class="mathCode">$c\in \{1,0,-1\}$. In this setting, we show that such a submanifold class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306291&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306291&_rdoc=1&_issn=0022247X&md5=24c0aadbae874e4aaa7973721c8db4df" title="Click to view the MathML source">Mⁿclass="mathContainer hidden">class="mathCode">$M^n$ must be either totally umbilical or isometric to a Clifford torus class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306291&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306291&_rdoc=1&_issn=0022247X&md5=45e952c267031d5ba0bff52826c2248f">class="imgLazyJSB inlineImage" height="20" width="165" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306291-si4.gif">class="mathContainer hidden">class="mathCode">${\mathbb{S}}^1\left(\sqrt{1-r^2}\right)\times {\mathbb{S}}^{n-1}(r)$, when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306291&_mathId=si23.gif&_user=111111111&_pii=S0022247X16306291&_rdoc=1&_issn=0022247X&md5=7093a35163339d2a3eae580756d9358f" title="Click to view the MathML source">c=1class="mathContainer hidden">class="mathCode">$c=1$, a circular cylinder class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306291&_mathId=si6.gif&_user=111111111&_pii=S0022247X16306291&_rdoc=1&_issn=0022247X&md5=edf33505c16a6f0a6158e02a080b0d6d" title="Click to view the MathML source">R×Sⁿ⁻¹(r)class="mathContainer hidden">class="mathCode">$\mathbb{R}\times {\mathbb{S}}^{n-1}(r)$, when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306291&_mathId=si32.gif&_user=111111111&_pii=S0022247X16306291&_rdoc=1&_issn=0022247X&md5=1a68b6514c1848b466bd38deb8850255" title="Click to view the MathML source">c=0class="mathContainer hidden">class="mathCode">$c=0$, or a hyperbolic cylinder class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306291&_mathId=si8.gif&_user=111111111&_pii=S0022247X16306291&_rdoc=1&_issn=0022247X&md5=dba62e31d95b3c62b6ddfd5666a28383">class="imgLazyJSB inlineImage" height="20" width="182" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306291-si8.gif">class="mathContainer hidden">class="mathCode">${\mathbb{H}}^1(-\sqrt{1+r^2})\times {\mathbb{S}}^{n-1}(r)$, when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306291&_mathId=si34.gif&_user=111111111&_pii=S0022247X16306291&_rdoc=1&_issn=0022247X&md5=4c8f5d7421a3442df706f82f06362a7e" title="Click to view the MathML source">c=−1class="mathContainer hidden">class="mathCode">$c=-1$. This characterization theorem corresponds to a natural improvement of previous ones due to Alías, García-Martínez and Rigoli br0020">[2], Cheng br0040">[4] and Guo and Li br0060">[6].