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.

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