Curvature identities derived from the integral formula for the first Pontrjagin number

Transplanting geometrical structures

“…Then, from the Theorem 1.2, it follows that Equations (3.d) and (3.e) also hold on any 4-dimensional almost Hermitian manifold which is not necessarily compact. This answers affirmatively questions posed in [13,26]. In particular, if M = (M, g, J) is a 4-dimensional Kähler manifold one has the following curvature identities [13,26]:…”

“…We also refer to related work of Euh, Jeong, and Park [9]. It follows from Theorem 1.2 that any such identity which holds in the compact setting automatically extends to the non-compact setting thus answering directly a question originally posed in [13,26].…”

“…In this section, we use Theorem 1.2 to answer certain questions that were raised previously in [13,26].…”

“…Euh, Park, and Sekigawa [13] derived the following curvature identities on M from the integral formula for the first Pontrjagin number using Novikov's theorem [29] provided that the underlying manifold M in question was compact:…”

“…Such problems arise in many contexts. One can often establish universal curvature identities for compact manifolds by considering the Euler-Lagrange equations of certain characteristic classes -see, for example, results in [10,11,12,13,17,26]. One then wants to show these identities hold more generally and this involves a transplanting problem.…”