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"> 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"> 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">. 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"> 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">, 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">, 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">, 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">, 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">, 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">. This characterization theorem corresponds to a natural improvement of previous ones due to Alías, García-Martínez and Rigoli
[2], Cheng
[4] and Guo and Li
[6].