# Relation with Jacobi fields in a small neighbourhood of some point in a complete manifold

I have the following question:

Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a sufficiently small neigbourhood of $p$ (one can assume that this neigbourhood is actually given by normal coordinates, hence by the exponential map $exp_{p}$). Let $\gamma : [0, \epsilon] \rightarrow M$ be a geodesic in this neighbourood starting at $p$. Let furthermore $J$ be a Jacobi field along with $J(0) = 0$ and $\frac{D}{dt}|_{t=0}J = v$ and $J(\epsilon) = w \not= 0$ and orthogonal to $\gamma$.

Can one then make the following approximation?

$$\frac{d(\gamma(t))}{2} \cdot \frac{d}{dt}|J|^{2} \geq |J|^{2}$$

If yes, why is this so?

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What do you mean ? –  pascal Jun 10 '12 at 9:31
Do you have an idea of this ? –  pascal Jun 10 '12 at 9:42
One can rephrease this in showing that: $d(\gamma(t)) \cdot g(\frac{D}{dt}J,J) \geq |J|^{2}$? Here any idea ? –  pascal Jun 10 '12 at 9:51