graphs of homomorphism of $(\mathbb{R}^{n},+)$ are minimal submanifolds. The same holds for ($S^{1},\times$). Are there generalizations of these statements?
2026-04-01 10:29:05.1775039345
Metric properties of graphs of lie groups homomorphisms
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Let $G$ be a Lie group equipped with a biinvariant semi-Riemannian metric. (For instance, take any compact Lie group with a biinvariant Riemannian metric.) Then nonconstant geodesics in $G$ are $G$-translates of 1-parameter subgroups of $G$. Accordingly, Lie subgroups of $G$ are totally geodesic. Now take the direct product of two such metrized groups $G\times H$ with the product semi-Riemannian metric. Graphs of continuous homomorphisms $G\to H$ are totally geodesic, hence, minimal, submanifolds of this product.