arXiv subjects

Sixuan Gu

Publications and source records attributed to Sixuan Gu.

2 recordsLinked to original sources

arXiv2026

A complete classification of left-invariant Einstein metrics on $S^3\times S^3$

We complete the classification of compact simply connected homogeneous Einstein manifolds in dimension six by resolving the remaining cases for left-invariant Einstein metrics on $G=\mathrm{SU}(2)\times\mathrm{SU}(2)\cong S^3\times S^3$. Previous work leaves two cases for the isotropy group $K$, namely $K=\{e\}$ and $K\cong\mathbb Z_2$. We first rule out $K=\{e\}$. Then we show that any $\mathbb Z_2$ symmetry generated by an inner involution $\sigma$ with $\tr\sigma=-2$ necessarily extends to a $\mathbb Z_2\times\mathbb Z_2$ symmetry. Together with the previously known classification results, these theorems imply that every left-invariant Einstein metric on $G$ is, up to homothety and isometry, either the standard product metric $g_{\rm can}$ or the Jensen nearly K\"ahler metric $g_{\rm NK}$.

math.DG↗

arXiv2026

Transpose Symmetry of Injectivity over Commutative Semirings

Let R be a commutative semiring, not necessarily with a multiplicative identity, and let A be an element of Mn(R). We prove that the map x to Ax on Rn is injective if and only if x to AT x is injective. Equivalently, the left- and right-cancellative elements of the multiplicative semigroup Mn(R) coincide. The proof splits formal determinant expansions into their even and odd halves; it uses no subtraction, additive cancellation, group completion, inverse, or multiplicative identity. As consequences we recover the stable-finiteness theorem for matrices over unital commutative semirings. We also prove that surjectivity is invariant under transpose. In fact, the existence of a surjective square matrix of positive size forces R to have a multiplicative identity.

math.RA↗