[Paper Review] On the curvature of invariant Kropina metrics

Hamid Reza Salimi Moghaddam|arXiv (Cornell University)|May 1, 2013

Advanced Differential Geometry Research8 references3 citations

TL;DR

This paper derives explicit formulas for the flag curvature of invariant Kropina metrics on homogeneous spaces, particularly focusing on naturally reductive and bi-invariant cases. By leveraging the Chern connection's coincidence with the Levi-Civita connection and applying Püttmann’s curvature tensor formula, the authors obtain closed-form expressions for flag curvature in terms of Lie algebra structures and inner products, significantly simplifying computation for these Finsler metrics.

ABSTRACT

In the present article we compute the flag curvature of a special type of invariant Kropina metrics on homogeneous spaces.

Motivation & Objective

• To compute the flag curvature of invariant Kropina metrics on compact homogeneous spaces G/H.
• To simplify the general flag curvature computation under the condition that the Chern connection of the Kropina metric coincides with the Levi-Civita connection of the underlying Riemannian metric.
• To derive specialized curvature formulas for naturally reductive and bi-invariant cases, reducing complexity in symmetric settings.
• To provide explicit, computable expressions for flag curvature using Lie algebraic data and inner products.

Proposed method

• Uses the Chern connection of the Kropina metric F = α²/β, assuming it coincides with the Levi-Civita connection of the underlying Riemannian metric g.
• Applies Püttmann’s curvature tensor formula for invariant metrics on G/H, adapted with a sign correction for consistency with the paper’s curvature definition.
• Expresses the flag curvature K(P,Y) via inner products and Lie bracket projections in the Lie algebra 𝔤 = 𝔥 ⊕ 𝔪.
• Derives g_Y(U,V) for the Kropina metric using the metric g and the vector field X corresponding to the 1-form β.
• Reduces the general formula to naturally reductive spaces by exploiting the condition B(X,[Z,Y]_𝔪) + B([Z,X]_𝔪,Y) = 0.
• Applies the bi-invariance property to simplify R(U,Y)Y = –1/4[[U,Y],Y] in the final formula.

Experimental results

Research questions

• RQ1 What is the explicit formula for the flag curvature of an invariant Kropina metric on a homogeneous space G/H when the Chern connection matches the Levi-Civita connection of g?
• RQ2 How does the flag curvature formula simplify in the case of naturally reductive homogeneous spaces?
• RQ3 Can the flag curvature of a Kropina metric defined on a Lie group with a bi-invariant metric be expressed in a more compact form?
• RQ4 What role does the positive definite endomorphism φ play in the curvature computation for invariant Kropina metrics?
• RQ5 How do the Lie algebra components and projections onto 𝔪 influence the flag curvature expression?

Key findings

• The flag curvature of an invariant Kropina metric on G/H is given by K(P,Y) = [3<U,X><R(U,Y)Y,X> + 2<Y,X><R(U,Y)Y,U>] / [2(<U,X>/<Y,X>)² + 2], where R(U,Y)Y is expressed via Lie brackets and φ.
• For naturally reductive spaces, the curvature simplifies to R(U,Y)Y = 1/4[Y,[U,Y]_𝔪]_𝔪 + [Y,[U,Y]_𝔥], enabling a more tractable curvature expression.
• In the bi-invariant case, the curvature tensor reduces to R(U,Y)Y = –1/4[[U,Y],Y], leading to the simplified flag curvature formula involving [[U,Y],Y].
• The formula for g_Y(U,V) is derived explicitly in terms of g(Y,X), g(U,X), and g(Y,Y), enabling curvature computation via the Finsler metric structure.
• The derivation confirms that the Chern connection coinciding with the Levi-Civita connection allows direct use of Riemannian curvature formulas in Finsler geometry.
• The final curvature formula in the bi-invariant case is K(P,Y) = [–3<U,X><[[U,Y],Y],X> – 2<Y,X><[[U,Y],Y],U>] / [8(<U,X>/<Y,X>)² + 8], providing a compact, computable expression.