Suppose is a complete Riemannian manifold. define . Obviously is continuous, what can we say about the smoothness of .

(1) is not near ;

(2) If is compact, then is not in

It seems that is not so smooth, let us consider

(3) is smooth at neighborhood of and is positive definite in .

(4) If is simply connected complete manifold with , then is on whole and is positive definite on .

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