| 总结: | <p>In the context of Hermitian geometry, the Hull--Strominger system is a system of non-linear PDEs on heterotic string theory, over a six-dimensional manifold endowed with an SU(3)-structure. Its seven-dimensional analogue, the heterotic <em>G</em><sub>2</sub> system, is a system for both geometric fields and gauge fields over a manifold with a <em>G</em><sub>2</sub>-structure. In this thesis, we study manifolds with geometric structures compatible with the Hull--Strominger system and the heterotic <em>G</em><sub>2</sub> system in the cohomogeneity one setting. In the former case, we develop a case-by-case analysis to provide a non-existence result for balanced non-Kähler SU(3)-structures which are invariant under a cohomogeneity one action on a simply connected six-manifold. In the latter case, we study two different SU(2)<sup>2</sup>-invariant cohomogeneity one manifolds, one non-compact <em>M</em> = <em>R</em><sup>4</sup> x <em>S</em><sup>3</sup>, and one compact <em>M</em> = <em>S</em><sup>4</sup> x <em>S</em><sup>3</sup>. For <em>R</em><sup>4</sup> x <em>S</em><sup>3</sup>, we prove the existence of a family of coclosed (but not necessarily torsion-free) <em>G</em><sub>2</sub>-structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed <em>G</em><sub>2</sub>-structure constructed from a half-flat SU(3)-structure is in this family. For <em>S</em><sup>4</sup> x <em>S</em><sup>3</sup>, we prove that there are no SU(2)<sup>2</sup>-invariant coclosed <em>G</em><sub>2</sub>-structures constructed from half-flat SU(3)-structures. Then, we study the existence of SU(2)<sup>2</sup>-invariant <em>G</em><sub>2</sub>-instantons on <em>R</em><sup>4</sup> x <em>S</em><sup>3</sup> manifold with the coclosed <em>G</em><sub>2</sub>-structures found. We find two 1-parameter families of smooth SU(2)<sup>3</sup>-invariant <em>G</em><sub>2</sub>-instantons with gauge group SU(2) on <em>R</em><sup>4</sup> x <em>S</em><sup>3</sup> and study its ``bubbling'' behaviour. We also provide existence results for locally defined SU(2)<sup>2</sup>-invariant <em>G</em><sub>2</sub>-instantons.</p>
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