Abstract We consider two different $$\text {SU}(2)^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SU</mml:mtext> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> -invariant cohomogeneity one manifolds, one non-compact $$M=\mathbb {R}^4 \times S^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> and one compact $$M=S^4 \times S^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> , and study the existence of coclosed $$\text {SU}(2)^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SU</mml:mtext> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> -invariant $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -structures constructed from half-flat $$\text {SU}(3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SU</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -structures. For $$\mathbb {R}^4 \times S^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> , we prove the existence of a family of coclosed (but not necessarily torsion-free) $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -structure constructed from a half-flat $$\text {SU}(3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SU</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -structure is in this family. For $$S^4 \times S^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> , we prove that there are no $$\text {SU}(2)^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SU</mml:mtext> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> -invariant coclosed $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -structures constructed from half-flat $$\text {SU}(3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SU</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -structures.

Izar Alonso

Annals of Global Analysis and Geometry

该杂志探讨了几何和分析的全球问题，以及这些领域之间的相互作用及其在理论物理问题中的应用。它有助于扩大该领域研究成果的国际交流。ANNALS OF GLOBAL ANALYSIS AND GEOMETRY 涵盖的领域包括：全局分析、微分几何、复流形以及复分析和代数几何的相关结果、李群、李变换群和谐波分析、变分微积分、微分几何和全局分析在理论物理问题中的应用。

The Annals of Global Analysis and Geometry addresses global issues in the fields of geometry and analysis, as well as the interactions between these areas and their applications in theoretical physics. The journal aims to enhance international communication of research findings in this field. Topics covered include global analysis, differential geometry, complex manifolds and related results in complex analysis and algebraic geometry, Lie groups, Lie transformation groups, harmonic analysis, variational calculus, and the applications of differential geometry and global analysis in theoretical physics.

《Annals of Global Analysis and Geometry》探讨几何和分析领域的全球性问题，以及这些领域之间的相互作用及其在理论物理中的应用。该杂志旨在促进国际间在这一领域研究成果的交流。涵盖的主题包括全局分析、微分几何、复流形及其相关的复分析和代数几何结果、李群、李变换群、谐波分析、变分微积分，以及微分几何和全局分析在理论物理问题中的应用。

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Coclosed $$G_2$$-structures on $$\text {SU}(2)^2$$-invariant cohomogeneity one manifolds

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