Abstract

In this thesis, the CR Bochner identity for energy density of a pseudoharmonic map is obtained in chapter 2, and we are able to find some interesting applications.

In chapter 3, we derive the CR Weitzenböck formula for 1-forms. By applying this formula, we are able to obtain a vanishing theorem.

In chapter 4, we derive the second variational formula for pseudoharmonic maps and solve a conjecture ([DT]) that any stable horizontal pseudoharmonic map φ from the pseudohermitian sphere S^3 into any Riemannian manifold N^m must be a constant map.

In chapter 5, we first derive the CR Lichnerowicz formula for a pseudohermitian spin^c manifold (M^(2n+1), J, θ). By a conformal transformation θ ̂=e^2u θ, where u is a positive real smooth function, we are able to have a lower bound for the first eigenvalue λ of the Dirac operator D_H on a 3-dimensional, closed pseudohermitian spin^c manifold. The uniform lower bound is 4μ_1, where μ_1 is the first eigenvalue
of the CR Yamabe operator.

In chapter 6, we will consider the product space H_1 × H_1, where H1 denotes the Heisenberg group, which is an example of pseudohermitian manifolds. We will investigate the geodesics and find the heat kernel on the product space H_1 × H_1.

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