Right-Invariant Riemannian Manifold

• Right-invariant Riemannian manifolds are differentiable spaces with metrics unchanged by right Lie group translations, ensuring uniform geometric behavior.
• They simplify the derivation of geodesic equations and Euler–Arnold dynamics, underpinning applications in hydrodynamics, nonlinear elasticity, and computational geometry.
• Their framework supports explicit metric constructions in both finite and infinite dimensions, advancing methods in Riemannian optimization and analysis.

A right-invariant Riemannian manifold is a differentiable manifold $M$ endowed with a Riemannian metric $g$ and a (right) Lie group action such that the metric is invariant under the right-translations induced by the action. This property is central to the metric and geometric theory of Lie groups and their homogeneous spaces, underpinning the formulation of geodesic equations, minimization problems, and applications in analysis, geometry, and applied mathematics.

1. Definition and Basic Properties

Let $G$ be a (finite- or infinite-dimensional) Lie group with identity element $e$, and let $\mathfrak{g}=T_eG$ denote its Lie algebra. A Riemannian metric $g$ on $G$ is called right-invariant if, for all $g \in G$, $v,w \in T_gG$, and for any $k \in G$,

$$g_{gk}\big( (dR_k)_g v, (dR_k)_g w \big) = g_g(v, w),$$

where $R_k: G \to G$, $R_k(h) = h k$, is right translation and $(dR_k)_g$ is its differential at $g$ (Larotonda, 2018, Bauer et al., 2018, Jensen et al., 2024).

Similarly, a metric can be left-invariant with the analogous property under left translations. At the identity, the metric is determined by a positive definite inner product $\langle \cdot, \cdot \rangle_e$ on $\mathfrak{g}$. For right-invariant metrics, the metric at arbitrary $g$ is given by

$$g_g(v, w) = \langle (dR_{g^{-1}})_g v, (dR_{g^{-1}})_g w \rangle_e.$$

This framework can be extended to homogeneous spaces $M=G/H$ via the natural projection $\pi: G \to G/H$, leading to induced (quotient) metrics on $M$ that are right-invariant under the (right) action of $G$ (Larotonda, 2018, Jensen et al., 2024).

2. Construction of Right-Invariant Metrics

The structure of right-invariant metrics depends crucially on the algebraic and representation-theoretic properties of $G$ and its subgroups. For matrix groups such as $\mathrm{GL}(n)$, a prominent example is the left-invariant, right-$O(n)$-invariant metric,

$$g_{CA}(CM, CN) = g_A(M, N),\quad \forall C,A \in \mathrm{GL}(n),~M,N \in T_A\mathrm{GL}(n),$$

$$g_{AQ}(MQ, NQ) = g_A(M, N),\quad \forall Q \in O(n),$$

where $O(n)$ is the orthogonal group (Martin et al., 2014). This implies that the inner product at the identity is $O(n)$-invariant. By representation theory, such an inner product on $\mathbb{R}^{n \times n}$ is parametrized via three coefficients $(\mu,\mu_c,\kappa)$:

$$\langle M, N \rangle_* = \mu \langle \operatorname{dev\, sym} M, \operatorname{dev\, sym} N \rangle +\mu_c \langle \operatorname{skew} M, \operatorname{skew} N \rangle +\frac{\kappa}{n} (\operatorname{tr} M) (\operatorname{tr} N),$$

where $\langle \cdot, \cdot \rangle$ is the Frobenius product, and $$\operatorname{sym}, \operatorname{skew}, \operatorname{dev}$$ denote the symmetric, skew-symmetric, and traceless projections, respectively.

On infinite-dimensional groups such as $\mathrm{Diff}(S^1)$ or $\mathrm{Diff}(M)$, right-invariant metrics can be constructed by defining an inner product on the Lie algebra of vector fields (using, e.g., a Sobolev $H^k$ metric or more generally a Fourier multiplier), then pushing it forward via right translation (Escher et al., 2012, Bauer et al., 2018). The metric is typically weak in the analytic sense:

$$G_\varphi(U, V) = (U \circ \varphi^{-1}, V \circ \varphi^{-1})_A,$$

where $A$ is a symmetric, positive-definite (pseudo-)differential or Fourier multiplier operator.

3. Geodesics, Exponential Maps, and the Euler–Arnold Equation

The geodesic equation on a right-invariant Riemannian manifold—especially a Lie group—is determined by the Levi-Civita connection, frequently computed via the Koszul formula. For right-invariant metrics, the connection on right-invariant vector fields significantly simplifies:

$$2g(\nabla_X Y, Z) = g([X,Y],Z) - g([Y,Z],X) + g([Z,X],Y),$$

where $[\cdot,\cdot]$ is the Lie bracket (Jensen et al., 2024).

For bi-invariant metrics, the connection is $\nabla_X Y = \frac{1}{2}[X, Y]$, and the geodesic equation $\nabla_{\dot\gamma}\dot\gamma = 0$ reduces to the familiar one-parameter subgroups $\gamma(t) = \exp(tX)$ being geodesics (Larotonda, 2018, Martin et al., 2014).

In the right-invariant (not bi-invariant) case, the geodesic flow on the group $G$ translates to the Euler–Arnold equation on the Lie algebra:

$$\frac{d}{dt} \mu(t) + \operatorname{ad}^*_{u(t)} \mu(t) = 0, \qquad \mu(t) = A u(t),$$

where $A$ is the inertia operator, and $\operatorname{ad}^*$ is the coadjoint representation. This covers a wide class of PDEs in hydrodynamics, including the Euler, KdV, and Camassa–Holm equations (Bauer et al., 2018, Escher et al., 2012).

Explicit formulas for geodesics in matrix groups with left-invariant, right-$O(n)$-invariant metrics are available, e.g.,

$$A(t) = A_0 \exp(t [\operatorname{sym} M - \frac{\mu_c}{\mu} \operatorname{skew} M]) \exp(t [1 + \frac{\mu_c}{\mu}] \operatorname{skew} M),$$

which describes all minimizing geodesics connecting $A_0$ to $A_1$ in $\mathrm{GL}^+(n)$ (Martin et al., 2014).

4. Geometric and Analytic Aspects in Infinite Dimensions

For infinite-dimensional Lie groups, such as diffeomorphism groups, right-invariant Riemannian metrics of Sobolev (or fractional Sobolev) type play a central role in geometric analysis and the theory of hydrodynamic PDEs. The key features are:

• The corresponding Euler–Arnold equation, interpreted as the geodesic equation for the right-invariant metric, translates to a nonlinear PDE on vector fields (e.g., the Burgers, Camassa–Holm, Hunter–Saxton, Euler–Weil–Petersson models) (Escher et al., 2012, Bauer et al., 2018).
• Well-posedness of geodesics is tightly linked to the analytic properties of the inertia operator $A$. For $H^s$-type metrics, global well-posedness may be established for $s\ge1/2$, and local existence for $s$ tailored to the regularity of the function spaces (Escher et al., 2012).
• The Riemannian exponential map at the identity is a smooth local diffeomorphism under suitable symbol conditions on $A$ (Escher et al., 2012).
• The "no loss, no gain" regularity principle holds: if initial data for the geodesic flow lies in $H^{q+1}$, solutions remain in $H^{q+1}$ for the existence interval.

In practical and physically-motivated scenarios, only invariance under a subgroup may be available (semi-invariance), as occurs with density-dependent metrics in shallow water models and compressible hydrodynamics (Bauer et al., 2018). For semi-invariant metrics, the extension of existence and persistence results requires higher regularity and additional structural assumptions.

5. Induced Geometry on Homogeneous Spaces

Given a Lie group $G$ with a (right-)invariant Riemannian metric and a closed subgroup $K \subset G$, a natural Riemannian metric on the homogeneous space $M = G/K$ arises via quotient construction (Larotonda, 2018, Jensen et al., 2024). The induced distance can be characterized by

$$d_M(g\cdot p, h\cdot p) = \inf_{k \in K} d_G(g, h k),$$

and the quotient norm on $T_{g\cdot p} M$ by minimizing among lifts differing by elements of the Lie algebra $\mathfrak{k}$.

Properly, the induced metric is a $G$-invariant Riemannian submersion metric, with explicit forms computable in finite-dimensional cases (as for Stiefel or Grassmann manifolds, symplectic or orthogonal group quotients) and in infinite-dimensional analogues (e.g., spaces of densities or universal Teichmüller space) (Martin et al., 2014, Jensen et al., 2024).

6. Applications and Computational Aspects

Right-invariant Riemannian manifolds underpin a range of applications:

• Nonlinear elasticity: The squared geodesic distance from $F$ to $SO(n)$, calculated under a left-invariant, right-$O(n)$-invariant metric on $\mathrm{GL}^+(n)$, coincides with the Hencky strain energy, identifying the geometry as the natural setting for isotropic hyperelastic energies (Martin et al., 2014).
• Hydrodynamics and PDEs: Right-invariant Sobolev metrics on $\mathrm{Diff}(M)$ induce Euler–Arnold equations modeling incompressible/compressible fluids, shallow water waves, and ideal fluid flows (Bauer et al., 2018).
• Riemannian optimization: Algorithms for optimization on manifolds such as the symplectic Stiefel manifold rely on explicit right-invariant metric structures, enabling matrix formulas for gradients, Hessians, and retractions efficiently tailored for numerical computation (Jensen et al., 2024).
• Metric geometry of operator spaces: Right-invariant metrics on infinite-dimensional groups, e.g., groups of invertible operators or diffeomorphism groups, provide a setting for the analysis of minimal geodesics and metric completeness in Banach and Hilbert manifolds (Larotonda, 2018).

A plausible implication is that right-invariant Riemannian structures provide a powerful and flexible geometric framework for both analysis and computation across finite and infinite dimensions.

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References:

• (Martin et al., 2014): "Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics"
• (Escher et al., 2012): "Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle"
• (Larotonda, 2018): "The metric geometry of infinite dimensional Lie groups and their homogeneous spaces"
• (Jensen et al., 2024): "Riemannian optimization on the symplectic Stiefel manifold using second-order information"
• (Bauer et al., 2018): "Semi-invariant Riemannian metrics in hydrodynamics"