Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question

This is simply the triangle inequality:

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

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.