# Sufficient condition on extending a vector to a parallel vector field.

Extending a tangent vector to a parallel vector field

Say we have a vector $$Z \in T_pM$$ for Riemannian manifold $$(M,g)$$. Lets consider a possible method for extending this vector to a parallel vector field. Let us choose coordinates $$(x,y)$$ centered at $$p$$, and first parallel translate $$Z$$ along the $$x$$ axis, and then the $$y$$ axis.

Using the L-C connection, it follows that $$\triangledown_{\partial_y}Z=0$$ for all points in our coordinate chart and that $$\triangledown_{\partial_x}Z=0$$ when $$y=0$$. We now consider the question of whether $$\triangledown_{\partial_x}Z=0$$ for all points in the coorindate chart. By uniqueness of parallel translates it would suffice to show that $$\triangledown_{\partial_y} \triangledown_{\partial_x}Z =0$$.

I don't understand why this last sentence is true. I understand that parallel translates are unique but I don't see why that shows that $$\triangledown_{\partial_y} \triangledown_{\partial_x}Z =0 \rightarrow \triangledown_{\partial_x}Z=0$$

This post is based on a dicussion in John Lee's Introduction to Riemannian Manifolds Chapter 7 page 117

Let me give you a hint.

Can you give one vector field $$W$$ that satisfies the two conditions (a) $$W = 0$$ for $$y=0$$ and (b) $$\nabla_{\partial_y} W = 0$$ everywhere?

Then use the fact that a vector field satisfying these two conditions is unique. So any vector field satisfying the two conditions, must be equal to the vector field $$W$$ you found.

The last thing to realize is that in your question $$W = \nabla_{\partial_x}Z$$.

• ... Is $W$ the zero vector field?
– user637978
Commented May 9, 2020 at 19:12
• And why is a vector field satisfying those two conditions unique???
– user637978
Commented May 9, 2020 at 19:13
• Yes, $W$ is the zero vector field. And W is unique because the parallel translation of a vector field is unique. Indeed, the condition $\nabla_{\partial_y}W=0$ exactly means that $W$ is a parallel vector field along the coordinate curves $v \mapsto (x,v)$. Commented May 9, 2020 at 20:59