Let $(M,g)$ be an $n$-dimensional smooth Riemannian manifold and $p_0\in M$, $v_0\in T_{p_0}M$ with $|v_0|=1$. $\nabla$ is the Levi Civita connection.
How can we construct a smooth vector field $X$ on $M$ such that \begin{array}{l} &X(p_0)=v_0,\\ &g(X,X)\leq1\ \ \ \text{ on }M,\\ &\nabla X(p_0)=0\in T_{p_0}M\otimes T^\ast_{p_0}M,\ \text{and }\\ &\Delta X(p_0)=0\in T_{p_0}M ? \end{array} Thank you.
Let $\epsilon > 0$ be small enough that the exponential map is injective on $B_\epsilon(0) \subset T_{p_0} M$. Define a local vector field $V$ on $B_\epsilon(p_0)$ by parallel transporting $v_0$ along radial geodesics - you should be able to check that this satisfies the required conditions, with the caveat that it is not defined on all of $M$.
Then just take a bump function $\rho \in C^\infty (M,[0,1])$ that is $1$ inside $B_{\epsilon/3}(p_0)$ and $0$ outside $B_{\epsilon/2}(p_0)$ and define $X = \rho V$.