Let $v^i$ be a vector on a Riemannian 3-manifold with metric $g_{ij}$ embedded inside a 3+1 space-time such that for some constant $N_M$ it satisfies the inequality $g_{ij}v^iv^j \leq N_M ^2$. Let $K$ be a symmetric rank-2 tensor on the 3-manifold. Then apparently the following holds:
$$\vert K_{ij} v^i v^j \vert \leq \vert K \vert _g N_M ^2.$$
This looks like some sort of a Cauchy-Schwarz inequality but given that $K$ is a tensor as described, I don't understand what the notation on the RHS means. For a rank-2 symmetric tensor $K$ what does $\vert K \vert _g$ mean?
If one knows that for some function $N$, $g_{ij}v^iv^j \leq N^2,$ where the function $N$ is itself bounded between constants
$$N_m \leq N \leq N_M,$$
then using inequalities like the above one can apparently show the following bound:
$$\int _{t_1} ^t \frac{1}{N} \Big(-v^i \partial _i N – \frac{dN}{dt} + K_{ij}v^iv^j\Big) dt
\leq -2\log N_m + \frac{1}{N_m} \int _{t_1}^t (\vert \nabla N \vert _g N_M + \vert K \vert _g N_M ^2 )dt,$$
for some fixed $t_1$ and $t$.
I can't understand that first "log" term in the above.
Also once the above bound is shown does it follow that the integral can be unbounded above or below depending solely on the property of the function $N$? If yes then what would be needed of $N$ to make the integral unbounded above or below?
Best Answer
I am just going to answer the first question. For a two-tensor we have
$$|T|^2 = \left< T,T \right> = g^{ik}g^{jl}T_{ij}T_{kl},$$
where $g$ is the metric, and the summation convention is understood. Note that this is the standard inner product structure induced by $g$, as mentioned by Jason DeVito. You might like to try and prove that this really does define a norm in the Riemannian case.
More generally, for a $(k,l)$ ($k$ times contravariant, $l$ times covariant) tensor field $T$ (on the manifold) we have
$$|T|^2 = \left< T,T \right> = g^{j_1q_1}g^{j_2q_2}\cdots g^{j_lq_l}g_{i_1p_1}g_{i_2p_2}\cdots g_{i_kp_k}T^{i_1i_2\cdots i_k}_{j_1j_2\cdots j_l}T^{p_1p_2\cdots p_k}_{q_1q_2\cdots q_l}.$$
For your other questions, you could do worse than look in 'Einstein Manifolds' by Besse.