Bulletin of the Brazilian Mathematical Society, New Series | 2021
A Splitting Result for Real Submanifolds of a Kähler Manifold
Abstract
<jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$$(Z,\\omega )$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>Z</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>ω</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> be a connected Kähler manifold with an holomorphic action of the complex reductive Lie group <jats:inline-formula><jats:alternatives><jats:tex-math>$$U^\\mathbb {C}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:msup>\n <mml:mi>U</mml:mi>\n <mml:mi>C</mml:mi>\n </mml:msup>\n </mml:math></jats:alternatives></jats:inline-formula>, where <jats:italic>U</jats:italic> is a compact connected Lie group acting in a hamiltonian fashion. Let <jats:italic>G</jats:italic> be a closed compatible Lie group of <jats:inline-formula><jats:alternatives><jats:tex-math>$$U^\\mathbb {C}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:msup>\n <mml:mi>U</mml:mi>\n <mml:mi>C</mml:mi>\n </mml:msup>\n </mml:math></jats:alternatives></jats:inline-formula> and let <jats:italic>M</jats:italic> be a <jats:italic>G</jats:italic>-invariant connected submanifold of <jats:italic>Z</jats:italic>. Let <jats:inline-formula><jats:alternatives><jats:tex-math>$$x\\in M$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>x</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:mi>M</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>. If <jats:italic>G</jats:italic> is a real form of <jats:inline-formula><jats:alternatives><jats:tex-math>$$U^\\mathbb {C}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:msup>\n <mml:mi>U</mml:mi>\n <mml:mi>C</mml:mi>\n </mml:msup>\n </mml:math></jats:alternatives></jats:inline-formula>, we investigate conditions such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$G\\cdot x$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mo>·</mml:mo>\n <mml:mi>x</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> compact implies <jats:inline-formula><jats:alternatives><jats:tex-math>$$U^\\mathbb {C}\\cdot x$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:msup>\n <mml:mi>U</mml:mi>\n <mml:mi>C</mml:mi>\n </mml:msup>\n <mml:mo>·</mml:mo>\n <mml:mi>x</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> is compact as well. The vice-versa is also investigated. We also characterize <jats:italic>G</jats:italic>-invariant real submanifolds such that the norm-square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of <jats:inline-formula><jats:alternatives><jats:tex-math>$$(Z,\\omega )$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>Z</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>ω</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> generalizing a result proved in Gori and Podestà (Ann Global Anal Geom 26: 315–318, 2004), see also Bedulli and Gori (Results Math 47: 194–198, 2005), Biliotti (Bull Belg Math Soc Simon Stevin 16: 107–116 2009).</jats:p>