Abstract
An Einstein connection which is both a special connection and a (k)-connection is called anSE(k)-connection. And a generalized even-dimensional Riemannian manifoldXn with the so-called “SE(k)-condition” defined by theSE(k)-connection is called theSE(k)-manifold. We obtain the necessary and sufficient condition that there is a uniqueSE(k)-connection inXn. Next, using these results, we define theSE(k)-manifold and study the properties of the curvature tensors and the field equations in theSE(k)-manifoldXn.
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