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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?

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This is simply the triangle inequality:

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

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Thanks! I should have seen this. Very stupid of me. –  Anirbit Oct 14 '10 at 12:05
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