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In thispaper we shall study minimal submanifolds in compact symmetric spaces and homologically volume minimizing submanifolds in compact simple Lie groups and quanternionic Kahler manifolds. The firstsubjectis studied by computing the second fundamental forms of submanifolds. In Section 2 using the structure theorem of the firstconjugate loci of compact symmetric spaces (Takeuchi [5])we compute the second fundamental form of a certainsubmanifold which is open and dense in the firstconjugate locus of a compact symmetric space and prove the minimality of it. Moreover we show that the submanifold has no geodesic point. The second subjectis studied by using the notion "calibration" introduced by Harvey and Lawson [2]. This notionisused in Sections 3 and 4. The fundamental 2-form of a Kahler manifold is one of important examples of calibrations. It satisfiesWirtinger's inequality, which can be statedas follows. Let M be a Kahler manifold with fundamental 2-form o). Then

Certain minimal or homologically volume minimizing submanifolds in compact symmetric spaces