What is the meaning of a second order differential operator in differential geometry?

Question

We consider $\mathbb{R}^n$ as a manifold $M$. A first-order differential operator $c^i \partial_{x_i} : C^\infty (M) \to C^\infty (M)$ can be viewed as a vector field on $M$ with coordinates $c^i$. However, a second-order differential operator $c^{ij}\, \partial_{x_i x_j} : C^\infty (M) \to C^\infty (M)$ cannot, because it is not a derivation : $$ c^{ij}\, \partial_{x_i x_j} (fg) \neq c^{ij}\, \partial_{x_i x_j} (f)\cdot g + f\cdot c^{ij}\, \partial_{x_i x_j}(g).$$ Since it is not a vector field, to what object does $c^{ij}\, \partial_{x_i x_j}$ correspond in differential geometry ? 

Answer 1

A second order differential operator is in general not easily defined in terms of vector and tensor fields.

It is defined to be a linear operator $F: C^\infty(M) \rightarrow C^\infty(M)$ such that the following hold:

1. It is a local operator, which means that if if $u\in C^\infty(M)$ vanishes on an open $O\subset M$, then $F(u)$ vanishes on $O$ too. Equivalently, if $F(u)$ is nonzero an open $O \subset M$, then $u$ is nonzero on $O$.

2. Given any coordinate chart $O \subset M$ with coordinates $(x^1, \dots, x^n)$, there exist functions $c^{ij}, b^k, a$, where $1 \le i,j,k \le n$ such that for any $u \in C^\infty_0(O)$, $$ F(u) = c^{ij}\partial^2_{ij}u + b^k\partial_k u + a u. $$

If $M$ has a connection (such as the Levi-Civita connection of a Riemannian metric), then any section $\sigma$ of $S^2T^*M$ defines a 2nd order differential operator \begin{align*} F: C^\infty(M) &\rightarrow C^\infty(M)\\u &\mapsto \sigma(\nabla^2 u) .\end{align*} However, not every 2nd order differential operator can be written in this form.

Answer 2

A second order differential operator like $$\partial_{x,y} +b \partial_x$$ or $$\sum \left (g_{i,k}\partial_{x_i, x_k} + b_i \partial_{x_i} \right)$$ comes with a symmetric metric matrix $g$ and a volume density $\rho$ in bilinear integrals.

The metric can be of different type, elliptic (positive eigenvalues), or hyperbolic (mixed positive and negative), or parabolic (some zeros, but a first derivative in that special dimension).

The second order derivative expressions derives from the bilinear form on the tangents

$$ \int \rho\ (\nabla \ f) \cdot g \cdot \ (\nabla h )\ d^nx \quad \longrightarrow \quad - \int \rho\ f \ \left( \rho^{-1} \ \nabla \left(g \ \nabla f \right)\ \right) d^nx $$ by partial integration in spaces of fast descent or zero boundary integral values.

Since elliptic, hyperbolic and parabolic equation have completely different meanings, solution spaces and methods, they cover completely different areas of applications.

Elliptic systems may be considered as equilibrium states of parabolic systems: the first order derivative is with respect to time and the Laplacian is a smoothing operator, time evolution leads to a time independent equilibrium with zero Laplacian.

Hyperbolic systems with one negative and $n$ positive eigenvalues are wave operators, that yield dynamic equations of motion of points in time from a given configuration of field and its time derivative at time zero.

The stress of local deformation curvature yields a net force from the boundary of a ball to its content, yielding waves in space and ocillations in time.

Hyperbolic equations are the playing ground of field dynamics in mathematical physics.