Why should the diffusivity matrix (of elliptic operator) map tangent space to itself? I have seen that an elliptic operator $A$ on a hypersurface $\Gamma$, written as $$Au=-\nabla_\Gamma \cdot (M(x)\nabla_\Gamma u)$$
(where $\nabla_\Gamma$ is the tangential or surface gradient) is constrained to satisfying

The matrix $M(x)$ must map the tangent space at $x$ to itself again.

Why is this? 

So $M$ cannot in general be a constant matrix

Which surprises me somewhat.
 A: If we consider $\nabla_\Gamma$ to be the covariant derivative operator associated with the metric $g_\Gamma$ on the hypersurface $\Gamma$ induced from the metric $g$ in the ambient manifold, then for any two tangent vector fields $X$,$Y$ on $\Gamma$, $(\nabla_\Gamma)_X Y$ is also a vector field on $\Gamma$, and thus we may in addition view the map $\nabla_\Gamma Y:  X \to (\nabla_\Gamma)_X Y$ as a linear transformation acting on the tangent spaces of $\Gamma$.  The divergence of $\nabla_\Gamma \cdot Y$ of $Y$ with respect to the operator $\nabla_\Gamma$ may be defined, in a coordinate-independent manner, to be the trace of this map.  (This is one of many equivalent definitions of $\nabla_\Gamma \cdot Y$.)  As such, we see that $\nabla_{\Gamma} \cdot Y$ is only defined for vector fields $Y$ tangent to $\Gamma$; thus the range of $M(x)$ must at every point $x$ lie in $T_x\Gamma$, the tangent space to $\Gamma$ at $x$. 
Is $M(x)$ constant?  Well, that depends what we mean by "constant".  Certainly there are linear endomorhisms of the tangent bundle of $\Gamma$ which are covariantly constant with respect to $\nabla_\Gamma$; the identity map is perhaps the prime example.  And indeed, if we consider the identity endomorphism on the tangent bundle of the ambient space, we see that it provides an example which is constant in.the ambient space and also restricts to a constant map on $T\Gamma$.  But if we write $I_{\text{ambient}} = P(x) + (I_{\text{ambient}} - P(x))$, where $P(x)$ is the projection onto the tangent space $T_x\Gamma$  of $\Gamma$, we see that $P(x)$ is in general not constant when considered as a function of $x$, since tangent spaces of $\Gamma$ vary from point to point in general.  But $I_{\text{ambient}}$ itself doesn't change; only the $M(x)$ invariant subspace $\text{Im}P(x) = T_x\Gamma$ does.  On the other hand, if we take $L(x)$ to be any matrix which actually varies with $x$, then for $x \in \Gamma$, $P(x)L(x)$ maps $T_x \Gamma$ to itself; by composing such $L(x)$ with $P(x)$ to obtain $M(x) = P(x)L(x)$ examples of non-constant $M(x):T_x \Gamma \to T_x \Gamma$ may be obtained; indeed, it is easy to see that all such $M(x)$ are of this form.  And so it is clear that $M(x)$ may be constant, but need not be.  Indeed, taking $M(x)$ to be to the covariantly constant endomorphism $I_\Gamma$ yields
$Au=-\nabla_\Gamma \cdot \nabla_\Gamma u, \tag{1}$
i.e. the differential operator $A$ is the ordinary Laplacian on $\Gamma$ in this case.  But other, non-constant $M(x)$ may be introduced to address specific applications.
In closing, I would like to point out an example of an equation of the general form
$Au=-\nabla_\Gamma \cdot (M(x)\nabla_\Gamma u) \tag{2}$
in which $M(x)$ typically varies from point to point, and which has some geometrical content as well.  Let $R_\Gamma$ be the Ricci tensor field associated with the metric $g_\Gamma$; it is a symmetric tensor of type $(0, 2)$; that is, it may be construed at each point $x \in \Gamma$ as a bilinear map $R_\Gamma:T_x \Gamma \times T_x \Gamma \to \Bbb R$ with $R_\Gamma(x, y) = R_\Gamma(y, x)$ for $x, y \in T_x\Gamma$.  Corresponding to $R_\Gamma$ there is a tensor field of type $(1, 1)$, which I shall denote by $R_\Gamma^*$, such that $g_\Gamma(x, R_\Gamma^*(y)) = R_\Gamma(x, y)$ for all $x, y$.  $R_\Gamma^*$ may be construed as an endomorphism $R_\Gamma^*:T_x \Gamma \to T_x \Gamma$; thus we have a second order partial differential operator $A_{R_\Gamma^*}$ defined on functions $u$ on $\Gamma$ via
$A_{R_\Gamma^*}u = -\nabla_\Gamma \cdot (R_\Gamma^*(\nabla_\Gamma u)); \tag{3}$
I believe $A_{R_\Gamma^*}$ is elliptic in the event that $R_\Gamma$, or $R_\Gamma^*$, is definite, either positive or negative.  I also believe that this equation has some differential geometric content, but exactly what that may be, at the present moment I cannot say.  Equation (3) does, however, provide an example from the class of operators (2) in which $M(x) = (R_\Gamma^*)(x)$ will in general not be constant.
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
