Ngô Quốc Anh

September 9, 2016

Benefits of “complete” and “compact” for analysis on Riemannian manifolds

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 10:08

When working on Riemannian manifolds, it is commonly assumed that the manifold is complete and compact. (The case of non-compactness is also of interest too.) In this entry, let us review the role of completeness and compactness in this setting.

How important the completeness is? Let us recall that for given a Riemannian manifold $(M,g)$, what we have is a nice structure as well as an appropriate analysis on any tangent space $T_pM$. For a $C^1$-curve $\gamma : [a,b] \to M$ on $M$, the length of $\gamma$ is $\displaystyle L(\gamma) = \int_a^b \sqrt{g(\gamma (t)) \langle \partial_t \gamma \big|_t, \partial_t \gamma \big|_t\rangle} dt$

where $\partial_t \gamma\big|_t \in T_{\gamma (t)}M$ a tangent vector. (Note that by using curves, the tangent vector $\partial_t \gamma\big|_t$ is being understood as follows $\displaystyle \partial_t \gamma\big|_t (f) = (f \circ \gamma)'(t)$

for any differentiable function $f$ at $\gamma(t)$.) Length of piecewise $C^1$ curves can be defined as the sum of the lengths of its pieces. From this a distance on $M$ whose topology coincides with the old one on $M$ is given as follows $\displaystyle d_g(x,y) = \inf_\gamma L(\gamma)$

where the infimum is taken on all over the set of all piecewise $C^1$ curves connecting $x$ and $y$.

Now the role of the completeness of the metric space $(M, d_g)$ can be seen from the Hopf-Rinow theorem:

Theorem (Hopf-Rinow). Let $(M, g)$ be a smooth Riemannian manifold. The following assertions are equivalent:

1. the metric space $(M, d_g)$ is complete,
2. any closed-bounded subset of $M$ is compact,
3. there exists $x \in M$ for which $\exp_x$, is defined on the whole of $T_xM$, and
4. for any $x \in M$, $\exp_x$ is defined on the whole of $T_xM$.

Here $\exp_x$ is the exponential map at $x$, which is locally defined, that is, it only takes a small neighborhood of the origin at $T_xM$, to a neighborhood of $x$ in the manifold. Moreover, one gets that any of the above assertions implies that any two points in $M$ can be joined by a minimizing geodesic. Here, a curve $\gamma$ from $x$ to $y$ is said to be minimizing if $L(\gamma) = d_g(x, y).$

We now move to the role of the compactness of manifolds.

How the compactness plays its role? Regarding to geodesics, one can define a so-called injectivity radius of $(M, g)$ at some point $x$, denoted by $inj_{(M,g)} (x)$, as the largest positive real number $r$ for which any geodesic starting from $x$ and of length less than $r$ is minimizing. Globally, the injectivity radius $inj_{(M,g)}$ is defined as follows $inj_{(M,g)} = \inf_x inj_{(M,g)} (x).$

Theorem. One has that $inj_{(M,g)}> 0$ for compact manifolds, but it may be zero for a completenoncompact manifold.

The injectivity radius helps us to define a cut locus $Cut(x)$ of $x$ as a subset of $M$, which has measure zero, such that $inj_{(M,g)} (x) = d_g(x, Cut(x)).$

Then the map $\exp_x$ is a diffeomorphism from some star-shaped domain of $T_xM$ at $0$ onto $M-Cut(x)$. Note that this notion is important since it makes sense to take the gradient and Hessian of the distance function away from the cut locus and $x$.

The compactness of $M$ also has some advantages when working with Sobolev spaces. For instance, we have the following simple result.

Theorem. If $M$ is compact, the Sobolev space $W^{k,p}(M)$ does not depend on the metric.

This can be easily seen from the following example: On $\mathbb R^n$ we define $g$ the standard Euclidean and $\displaystyle \widetilde g = \frac 4{(1+|x|^2)^2} g$

the metric after stereographic projecting the standard metric on the sphere $\mathbb S^n$. Clearly the constant function $1$ belongs to the Sobolev space associated with $\widetilde g$ while it does not for the standard metric $g$. This answers why variational methods do not have a place when working with complete non-compact Riemannian manifolds.

To end this entry, I refer the reader to the book

Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities (Courant Lecture Notes)

by Professor Emmanuel Hebey, which is very interesting and worth reading. The text mentioned above is basically adapted from this book.