[Paper Review] Universal curvature identities and Euler Lagrange Formulas for Kaehler manifolds

Peter Gilkey, JeongHyeong Park+1 more|arXiv (Cornell University)|Nov 11, 2013

Geometry and complex manifolds14 references3 citations

TL;DR

This paper establishes universal curvature identities and Euler-Lagrange equations for Kähler manifolds by relating characteristic invariants derived from curvature contractions with powers of the Kähler form. It proves that the Euler-Lagrange equations for scalar invariants formed by pairing invariant polynomials with the Kähler form coincide with symmetric 2-tensor valued universal curvature identities in the Kähler setting, extending real-setting results to the complex context via restriction maps and geometric variational methods.

ABSTRACT

We relate certain universal curvature identities for Kaehler manifolds to the Euler-Lagrange equations of the scalar invariants which are defined by pairing characteristic forms with powers of the Kaehler form.

Motivation & Objective

• To extend universal curvature identities from real Riemannian geometry to the Kähler setting, where complex structures and Kähler forms impose additional symmetries.
• To derive the Euler-Lagrange equations for scalar invariants constructed by pairing invariant polynomials (in curvature) with powers of the Kähler form.
• To establish a precise correspondence between these Euler-Lagrange equations and symmetric 2-tensor valued universal curvature identities in the Kähler context.
• To prove that the Euler-Lagrange operator for such invariants coincides with a universal curvature identity via restriction maps and geometric variation techniques.
• To generalize previous results in real geometry to the Kähler setting by leveraging holomorphic structure, curvature symmetries, and characteristic classes.

Proposed method

• Uses the Newlander-Nirenberg theorem to define holomorphic structures and decompose tensor bundles into J-invariant and J-anti-invariant components.
• Applies the Kähler identity $ R(x,y,z,w) = R(x,y,Jz,Jw) $ to exploit the complex structure in curvature symmetries.
• Defines scalar invariants via pairing symmetric polynomials $ ext{Tr}_ u $ on curvature with $ rac{1}{m!} ar{ u}^m $, the volume form.
• Employs restriction maps $ r_{m,k} $ from $ rak{S}_{m,k} $ to $ rak{S}_{k,k} $ to relate invariants on manifolds of different complex dimensions.
• Applies variational calculus to one-parameter families of Kähler metrics $ ilde{g}_ u $, computing first variations of invariants to derive Euler-Lagrange equations.
• Uses the isomorphism between $ S_+^2M $ and $ igwedge_+^2M $ via the Kähler form $ ar{ u} $ to translate tensor identities into curvature invariants.

Experimental results

Research questions

• RQ1 How do universal curvature identities in the Kähler setting relate to the Euler-Lagrange equations of scalar invariants formed by pairing characteristic forms with powers of the Kähler form?
• RQ2 Can the Euler-Lagrange equations for Kähler invariants be expressed as symmetric 2-tensor valued universal curvature identities?
• RQ3 What is the role of restriction maps $ r_{m,k} $ in relating curvature invariants on manifolds of different complex dimensions?
• RQ4 How does the variation of the metric along a family $ ilde{ u}_ u $ affect the integral of curvature contractions with $ ar{ u}^k $?
• RQ5 Under what conditions does the vanishing of the first variation of a Kähler invariant imply a universal curvature identity?

Key findings

• The Euler-Lagrange equations for scalar invariants $ rac{1}{k!} ar{ u}^k ullet ilde{ u}_k(ar{ u}) $ coincide with symmetric 2-tensor valued universal curvature identities in the Kähler setting.
• The map $ ilde{ u}_{rak{Q},m,k} $ from $ rak{S}_{m,k} $ to $ rak{K}_{rak{Q},m,k} $ is an isomorphism, ensuring that every such invariant arises from a unique universal curvature identity.
• The first variation of the functional $ rac{1}{k!} ar{ u}^k ullet ilde{ u}_k(ar{ u}) $ along a metric deformation $ ilde{g}_ u $ vanishes if and only if the curvature satisfies the universal identity $ ilde{ u}_{rak{Q},m,k} ilde{ u}_k = ilde{ u}_{rak{Q},m,k} ilde{ u}_k $.
• The restriction map $ r_{k+1,k} $ preserves the vanishing of curvature identities, allowing reduction to lower-dimensional Kähler manifolds to test identities.
• The proof relies on constructing a one-parameter family $ ilde{ u}_ u $ on $ ilde{ u}^{k+1} imes ilde{ u}_ u^1 $, showing that the variation of the integral equals the integral over the base manifold, leading to a contradiction if identities differ.
• The key identity $ rac{1}{k!} ar{ u}^k ullet ilde{ u}_k(ar{ u}) = ext{Tr}(r_{k+1,k} ilde{ u}_k(ar{ u})) $ holds on complex dimension $ k $ manifolds, enabling reduction to lower dimensions.