• 대한수학회논문집
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• 제38권2호
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• pp.629-641
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• 2023

The aim of this paper is two-fold. First, we study the Chinea-Gonzalez class C₁₂ of almost contact metric manifolds and we discuss some fundamental properties. We show there is a one-to-one correspondence between C₁₂ and Kählerian structures. Secondly, we give some basic results for Riemannian curvature tensor of C₁₂-manifolds and then establish equivalent relations among 𝜑-sectional curvature. Concrete examples are given.

• 호남수학학술지
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• 제5권1호
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• pp.45-69
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• 1983

The hermitian structures on complex manifolds have been studied by several mathematicians ([1], [2], and [3]), and the Kähler structure on hermitian manifolds have been so much too ([6], [12], and [15]). There has been some gradual progress in studying the invariant forms on Grassmann manifolds ([17]). The purpose of this dissertation is to prove the Theorem 3.4 and the Theorem 4.7, with relation to the nature of complex Grassmann manifolds. In $\S$ 2. in order to prove the Theorem 4.7, which will be explicated further in $\S$ 4, the concepts of the hermitian structure, connection and curvature have been defined. and the characteristic nature about these were proved. (Proposition 2.3, 2.4, 2.9, 2.11, and 2.12) Two characteristics were proved in $\S$ 3. They are almost not proved before: particularly. we proved the Theorem 3.3 : $G_{k}(C^{n+k})=\frac{GL(n+k,C)}{GL(k,n,C)}=\frac{U(n+k)}{U(k){\times}U(n)}$ In $\S$ 4. we explained and proved the Theorem 4. 7 : i) Complex Grassmann manifolds are Kahlerian. ii) This Kähler form is $\pi$-fold of curvature form in hyperplane section bundle. Prior to this proof. some propositions and lemmas were proved at the same time. (Proposition 4.2, Lemma 4.3, Corollary 4.4 and Lemma 4.5).