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Let M be an n-dimensional, complete, connected and locally compact Riemannian manifold and g be its metric. Denote by Δ M the Laplacian on M.The Brownian motion on the Riemannian manifold M is defined to be the unique minimal diffusion process (X t , ζ, P x , x e M) associated to the Laplacian Δ M where ζ is the explosion time i.e. if ζ(ω) < +oo, lim.X r ί (0} = lonM;otherwise it will be called transient. It has been known that the Brownian motion on a compact Riemannian manifold is recurrent. See for example Mckean [7] page 99. In this paper we shall restrict our consideration to non compact case and clarify the relation between the recurrent and transient properties of the process X and geodesies, curvature of M.In the first section we shall summarize recurrence and transience of the Brownian motion on a rotationally symmetric Riemannian manifold (See Section 1 for the precise definition). Section 2 is devoted to the discussion for the general case. Some examples will be shown in the last section.

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Let M be an ra-dimensional, complete, connected and non compact Riemannian manifold and g be its metric. Δ M denotes the Laplacian on M.The Brownian motion on the Riemannian manifold M is defined to be the unique minimal diffusion process (X t , ζ, P x , xeM) associated with the Laplacian Δ M where ζ(ω) is the explosion time of X t (ω) i.e. if ζ(ώ) < + oo, then limXXω) = oo.In the previous paper [3], the author has discussed recurrence and transience of the Brownian motion X on M. This paper may be considered to be a continuation, in which the relation between explosions of the Brownian motion X and geodesies, curvature of the Riemannian manifold M will be investigated. It should be remarked that Yau [7] has given a sufficient condition for no explosion of the Brownian motion in terms of the Ricci curvature.Let us begin with the Brownian motion X° = (X°t, ζ°, P°X 9 x e M o ) on a model (Λf 0 , g 0 ) where the model (Λf 0 , g 0 ) is defined to be a Riemannian mani- (ii) j +~g oiry^drj'goisy-'ds = +00 . In order to prove the above theorems, we shall introduce the following notations. u°p(x) = u°p(r)for x = (r, ί) e M o .Thus w° e C°°([0, ^)) satisfies u°p(d(p, x)). Therefore following an argument similar to Yau [6], Appendix, we can obtain under the assumption (i) thatthen it holds thati.e. lim u°p(r) = 0 for every r ^ 0 .Hence it follows from the inequality proved above that limu p (x) = 0 for every xeM.Since σ p -> ζ as p -> + oo, we see that 0 = lim w/x) = £ye~ζ} for every x e M.Thus we can conclude Proof of Theorem 2.2. We first note that under the assumptions exp p maps T P (M) diffeomorphically onto goi) dr )Then it can be easily seen that v(r) £ exp {^(r)} for every r ^ 1 and so v(r) is bounded above from the assumption (ii) of Theorem 2.2. Now applying Itό's formula to the function e~ιϋ{x), we obtain from the above inequality that Set v(x) = v(d(p, x)). Then with the geodesic polar coordinates (r, Θ) and G(r, Θ) = Vdet(g ί:j )(r, θ) where g = g^dxidx^ we havefor each x e Σ p -Σ ι where τ^ω) = mΐ{t > 0\d(p, X t {ω)) <^ 1}. Letting p -> + 00, we have u(oo)JS β {e-S ζ < τj + E x {e-«, ζ > rj ^ P(x) .because σ^, -> ζ as ^ -> + 00. We shall showThen it is easy to see thatFurthermore Hessian comparison theorem [1] gives thatConsequently we can deduce by virtue of the maximum principle,i.e. Pxfa < σ p ) ^ ψ p (d(p, x) , Yau has shown that no explosion for the Brownian motion is possible if the Ricci curvature of M is bounded from below by a constant. We shall extend this result as follows. If for a fixed p e M and every minimal geodesic m(r)with positive constants C* ί = 1, 2, then no explosion for the Brownian motion X is possible.Proo/. In order to prove this, it is enough to show the existence of a model (M o , g 0 ) which satisfies the conditions (i) and (ii) in Theorem 2.1.Set Ko(r) = -C,r 2 -C 2 , re [0, +oo) and let g o (r) e C([0, +oo)) be the unique solution of the following Jacobi equation.