Partial connections and contact instantons on contact manifolds
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We study partial connections that are defined on a vector bundle E over a contact distribution H of a contact manifold (M2m+1; [zero with middle tilde]) by adapting the Rumin complex of the exterior derivative in a contact case. Full connections will be investigated in a new manner using the partial connection's point of view (One can view it as a reduction method). The alternative one to one correspondence between a full connection D and a partial connection D is introduced by linking with some B 2 EndE, i.e. D = D(D;B). For instance, we provide the new constructions of the Tanaka Webster connection and the Tanaka canonical connection through the suitable pair (D;B). The contact instanton equation and Hermitian-Einstein connection over a contact manifold are explored using the above correspondence. Consequently, we prove the existence of a solution of the B-inhomogeneous Yang-Mills equation D F = mB [zero with middle tilde]. This resembles the Tian instanton and Hermitian-Einstein connection over a Kahler manifold. For the applications, results of Dragomir-Urakawa in CR manifolds are covered into contact manifolds.
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Ph. D.