We present a criterion of local normal embedding of a semialgebraic (or definable in a polynomially bounded o-minimal structure) germ contained in R n in terms of orders of contact of arcs. Namely, we prove that a semialgebraic germ is normally embedded if and only if for any pair of arcs, coming to this point the inner order of contact is equal to the outer order of contact.the anonymous referee for his patience and extremely useful comments and corrections. Normally embedded setsLet X ⊂ R n be a connected semialgebraic set. We define an inner metric on X as follows: Let x, y ∈ X. The inner distance d X (x, y) is defined as the infimum of lengths of rectifiable arcs γ : [0, 1] → X such that γ(0) = x and γ(1) = y. Notice that for connected semialgebraic sets the inner metric is well-defined.1991 Mathematics Subject Classification. 14B05; 32S50 .

Abstract. The paper is devoted to relations between topological and metric properties of germs of real surfaces, obtained by analytic maps from R 2 to R 4 . We show that for a big class of such surfaces the normal embedding property implies the triviality of the knot, presenting the link of the surfaces. We also present some criteria of normal embedding in terms of the polar curves.

We show that the knot type of the link of a real analytic map germ with isolated singularity f : (R 2 , 0) → (R 4 , 0) is a complete invariant for C 0 -A -equivalence.Moreover, we also prove that isolated instability implies C 0 -finite determinacy, giving an explicit estimate for its degree. For the general case of real analytic map germs, f : (R n , 0) → (R p , 0) (n ≤ p), we use the Lojasiewicz exponent associated to the Mond's double point ideal I 2 (f ) to obtain some criteria of Lipschitz and analytic regularity.

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Let Pkfalse(n,pfalse) be the set of all real polynomial map germs f=false(f1,⋯,fpfalse):false(Rn,0false)→false(Rp,0false) with degree of f1,⋯,fp less than or equal to k∈N. The main result of this paper shows that the set of equivalence classes of Pkfalse(n,pfalse), with respect to multi‐scriptK‐bi‐Lipschitz equivalence, is finite.