Classification of calibrations on R^n for n≤7

In this thesis, our aim is to understand all possible calibrated submanifolds of R^n for n≤7. We start with basic definitions and tools that are used throughout this thesis. Then we study the general theory of calibrated geometries and give some interesting examples of them, following the historical development of the subject. After examining calibrations on R^n with dimensions one, two, n-1, and n-2 for arbitrary n, we focus on the nontrivial cases: classification of 3-dimensional calibrations on R^6 and R^7. On R^6, we have four types of calibrations, namely special Lagrangian, Kähler, single point and double point calibrations. On R^7 in addition to these four cases, we get six new types, which are called as associative, CP^2, double CP^1, S^1, S^2 and S^3 calibrations.