# An inequality about norms of vector fields on Riemannian manifolds

Let $g_{ij}$ be the components of a symmetric rank-2 positive definite tensor (metric on a Riemannian manifold). Let ${C^i}$ and ${ \beta ^i }$ be components of a vector field on it, the former of which is a functions of a variable $t$ and let $N$ and $M$ be two positive constants. Now let the following inequality hold,

$$\sqrt {g_{ij} \left( \frac{dC^i}{dt} + \beta ^i \right) \left( \frac{dC^j}{dt} + \beta ^j \right)} < N < M$$

And there exists a constant $B$ such that $\sqrt{g_{ij} \beta ^i \beta ^j}$ is uniformly bounded by $B$.

Then apparently the following is true,

$$\sqrt {g_{ij} \frac{dC^i}{dt} \frac{dC^j}{dt} } \leq N + \sqrt{g_{ij} \beta ^i \beta ^j} \leq M + B$$

How does this follow?

-

$$\lVert\frac{dC}{dt}\rVert = \lVert\frac{dC}{dt} + \beta - \beta\rVert \le \lVert\frac{dC}{dt} + \beta \rVert + \lVert-\beta\rVert < N + \lVert\beta\rVert < M + B$$