I am reading now the book The Ricci Flow in Riemannian Geometry by Ben Andrews and Christopher Hopper. It has been a while since I did not use pullback bundles and other objects, and they still look complicated to me.
I have trouble regarding sections 3.2.1 and 3.2.2. First let me introduce some notations and context:
- They consider a vector subbundle of $T(M \times I)$, where $I \subset \mathbb{R}$ is a time interval, consisting of vector tangents to the first factor $M$ in the sense that $(\pi_2)_* = 0$. On that bundle, a connection $\nabla$ is defined via the one on $M$ denoted by $^M\nabla$ augmented by its effect on the time component $\nabla_{\partial_t} u = [\partial_t, u]$ for vectors on this bundle, and extended by zero outside the bundle.
- We are given a smooth map between Riemannian manifolds $f : (M, g) \to (N, h)$ and consider its time dependent variant $f : M \times I \to N$. The map induces the pullback bundle $f^*TN$ over $M \times I$, and consider on it the pullback connection.
- The Energy $E$ is defined as the total density $$E(f) = \frac{1}{2}\int_M e(f) \: d\mu(g) = \frac{1}{2}\int_M\langle f_*, f_*\rangle \: d\mu(g)$$
- We use coordinates $\{x_i\}$ on $M \times I$ and $\{y^\alpha\}$ on $N$. We also use the indices to denote components. For instance in coordinates, we have $f_* = (f_*)_i^\alpha dx^i \otimes (\partial_\alpha)_f$.
I have a question regarding the motivation behind the gradient descent flow: By a direct computation as done in section 3.2.1, the authors show that $$ \frac{d}{dt} E(f) = -\int_M \langle f_*\partial_t, \Delta_{g,h}f\rangle \: d\mu(g). $$ Then they go on like this
Note that $f_*\partial_t$ is the variation of $f$. Hence the gradient of E, with respect to the inner product of $f^*TN$, is $-\Delta_{g,h}f$ and the gradient descent flow is $$ f_*\partial_t = \Delta_{g,h}f.$$
First, I do not understand why is the gradient of $E$ is $-\Delta_{g,h}f$. Second, how does this imply the following of the statement?
I have another question regarding the evolution of the energy density: The authors show that the evolution equation for $f_*$ is given by $$ (\Delta_{\partial_t} f_*)(\partial_i) = (\Delta_* f)(\partial_i) + g^{kl}R^N(f_*\partial_k, f_*\partial_i)(f_*\partial_l) - g^{kl}f_*(R^M(\partial_k, \partial_l)).$$ From that, they infer that $$\frac{d}{dt}e = \langle f_*, \Delta f_*\rangle + g^{kl}g^{ij}R^N(f_*\partial_k, f_*\partial_i, f_*\partial_l, f_*\partial_j) - g^{kl}h_f\left(f_*\left((^M\mathrm{Ric})_k^{\:\,m}\partial_p\right), f_*\partial_l\right). $$ Here is my attempt for the second term : By unravelling the definition for the scalar product, we get \begin{multline} g^{kl}g^{ij}(h_{\alpha\beta})_f(f_*)_i^\alpha[R^N(f_*\partial_k, f_*\cdot)(f_*\partial_l)]_j^\beta = g^{kl}g^{ij}h_f\left((f_*)_i,R^N(f_*\partial_k, f_*\partial_j)(f_*\partial_l)\right) = g^{kl}g^{ij}R^N(f_*\partial_k, f_*\partial_j, f_*\partial_l, f_*\partial_i).\end{multline} For the third term. Here is my attempt: By unravelling again, we get \begin{multline} g^{kl}g^{ij}(h_{\alpha\beta})_f (f_*)_i^\alpha[f_*(R^M(\partial_k, \cdot)\partial_l)]_j^\beta = g^{ij}h_f\left((f_*)_i, g^{kl}f_*((R^M)_{kjl}^{\:\:\:\:m}\partial_m\right) \\ = - g^{ij}h_f\left((f_*)_i, f_*(g^{kl}(R^M)_{jkl}^{\:\:\:\:m}\partial_m\right) = g^{ij}h_f\left((f_*)_i, f_*(g^{kl}(R^M)_{jk}\!\,^m_{\:\:\,l}\partial_m\right) = g^{ij}h_f\left((f_*)_i, f_*((^M\mathrm{Ric})_j^{\:\,m}\partial_m)\right) = g^{kl}h_f\left(f_*\partial_l, f_*((^M\mathrm{Ric})_k^{\:\,m}\partial_m)\right). \end{multline} Are those two computations correct?