Abstract ▼

Purpose & contents:
The major object for this study is to further develop our understanding about importance of curvature in differential geometry. The purposes of this project are to get a curvature identity which holds on an arbitrary -dimensional manifold and its applications, furthermore to

Purpose & contents:
The major object for this study is to further develop our understanding about importance of curvature in differential geometry. The purposes of this project are to get a curvature identity which holds on an arbitrary -dimensional manifold and its applications, furthermore to get the curvature condition for harmonicity and minimality of characteristic vector fields on the contact metric manifold.
Result:
Berger[2] got a curvature identity on a 4-dim compact Riemannian manifold derived from the generalized Gauss-Bonnet formula. We proved that the curvature identity holds on any 4-dimensional Riemannian manifold which is not necessarily compact.
1. We study the symmetric 2-form valued homogeneous invariants of order 4, which are given by the curvature and the covariant derivatives of the curvature tensor, to give a new proof of Labbi' s result concerning the Euler-Lagrange equations of the Gauss-Bonnet integral, and to give a new derivation of the identity of ours, and we generalized the identity to the higher dimensional setting, to the pseudo Riemannian setting, and to the Riemannian manifolds with boundary setting. From this research the word "Euh-Park-Sekigawa identity" is adopted as a keyword.
2. From the integral formula for the first Pontrjagin number and the first Chern number on a 4-dimensional compact almost Hermitian manifold, we derived curvature identities to hold on any 4-dimensional almost Hermitian manifold which is not necessarily compact and we extend the identity to the higher dimensional Kaehler manifold.
A contact metric manifold is said to be H-contact if the characteristic vector field is harmonic. An eta-Einstein manifold is an example of a H-contact manifold. A 3-dimensional H-contact manifold M is a generalization of Sasakian manifold on an open dense subset of M.
3. We studied harmonicity and minimality of characteristic vector fields on the contact metric manifold and also discussed the relationship between the generalization of Sasakian manifolds and eta-Einstein contact manifolds.
4. We extended a result of Patodi from the closed Riemannian setting to the context of closed contact setting. We proved that the compact eta-Einstein Sasakian manifold of dim ≥ 5 is spectrally determined.
5. Every algebraic model of the curvature tensor is geometrically realizable by a compact manifold.
Expected Contribution:
Curvature is a central tool in differential geometry. We can relate curvature to the underlying geometry and topology of a manifold, and examine the analytic properties which are influenced by curvature. The results obtained by curvature identities give a variety of useful information relating to geometry and multiple fields in mathematics. niversal symmetric 2-form valued homogeneous invariants play an important role for the proof of the Gauss-Bonnet theorem and Riemannian-Roch theorem for Kaehler manifolds using heat equation methods