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In this note we consider two methods in order to investigate volume minimizing submanifolds in compact symmetric spaces. The first is calibration ([4]) and the second is integral geometry. We can show that certain submanifolds are volume minimizing in their real homology classes using calibrations. A calibration is a closed differential form on a Riemannian manifold which satisfies a certain inequality. A definition of calibrations will be given in Section 1. On the othor hand we can prove that certain submanifolds are volume minimizing in its homotopy classes using integral geometry. We shall use a generalized Poincare's formula in Riemannian homogeneous spaces given by Howard [7].

Volume minimizing submanifolds in compact symmetric spaces