关于带左不变度量的矩阵李群测地线与单参数子群的关系及相关研究资料咨询

Hey there! Great question—this is a really fundamental topic in Riemannian geometry on matrix Lie groups, so let's break it down clearly.

Key Relationship: 测地线 vs 单参数子群 #

First off, the short answer is: they aren't always the same, but there's a tight connection depending on the metric.

• Special case: Bi-invariant metrics
  If your left-invariant metric is also right-invariant (called a bi-invariant metric), then geodesics starting at the identity element are exactly the one-parameter subgroups. For any other point in the group, geodesics are just left-translates of these one-parameter subgroups.
  This is true for compact semisimple Lie groups like SO(n) (rotations) or SU(n) (unitary matrices)—the Killing form (a natural inner product on the Lie algebra) induces a bi-invariant metric here, making the relationship super clean.

• General left-invariant metrics
  For arbitrary left-invariant metrics, geodesics don't have to be one-parameter subgroups. Instead, the connection is governed by the Levi-Civita联络 adapted to the left-invariant metric. Using the Koszul formula (which defines the Levi-Civita联络 for any inner product), you can derive that the geodesic equation on the Lie group translates to a differential equation on the Lie algebra (a special case of the Euler-Poincaré equation).
  A one-parameter subgroup generated by a Lie algebra element (X) will be a geodesic only if (X) satisfies a specific orthogonality condition: (\langle [Z, X], X \rangle = 0) for all (Z) in the Lie algebra (here (\langle \cdot, \cdot \rangle) is the inner product on the Lie algebra defining the left-invariant metric). In plain terms, this means (X) has to be "orthogonal" to all commutators it forms with other elements of the Lie algebra.

Examples to Illustrate #

• GL(n, ℝ) with standard left-invariant metric
  Take the inner product on (\mathfrak{gl}(n, ℝ)) (the Lie algebra of GL(n, ℝ)) as (\langle X, Y \rangle = \text{tr}(X^T Y)). This is left-invariant but not bi-invariant. For symmetric or skew-symmetric (X), the orthogonality condition above holds, so their one-parameter subgroups (like (\exp(tX)) where (X) is symmetric, which gives positive definite matrices) are geodesics. But for a non-symmetric, non-skew-symmetric (X) (e.g., (X = \begin{pmatrix}1 & 1 \ 0 & 0\end{pmatrix})), the condition fails—its one-parameter subgroup isn't a geodesic.

• SO(n) with bi-invariant metric
  On SO(n), the Killing form induces a bi-invariant metric. Every one-parameter subgroup is a rotation about a fixed axis (e.g., (\exp(tX)) where (X) is skew-symmetric), and these are exactly the geodesics starting at the identity. Any geodesic in SO(n) is just a left-translate of such a subgroup (i.e., rotating first then translating, or vice versa—since the metric is bi-invariant, left and right translations are isometries).

References for Further Study #

Here are some go-to resources at different levels:

• Introductory: Lie Groups, Lie Algebras, and Representations by Brian Hall. It has a dedicated chapter on Riemannian geometry of Lie groups, with clear explanations of left-invariant metrics, geodesics, and the bi-invariant case.
• Classic textbook: Riemannian Geometry of Lie Groups by Serge Lang. This is a deep dive into the topic, covering everything from basic definitions to advanced results on geodesics and curvature.
• Undergraduate-friendly: Differential Geometry of Curves and Surfaces by Manfredo do Carmo. The final chapter touches on Lie groups, including left-invariant metrics and simple geodesic examples.
• Advanced/applied: Geometric Control of Mechanical Systems by Bullo and Lewis. It connects Lie group geodesics to Euler-Poincaré equations, which are crucial for applications in robotics and mechanics.

Hope this gives you a solid starting point! If you want to dig into specific examples or equations, feel free to follow up.

备注：内容来源于stack exchange，提问作者gsoldier