Smooth surface or smooth metric? Confused!

Question

I am a student biginner in differential geometry, and i found a lot of times the term, "smooth" surface in articles of semi-Riemannian geometry. If we say that $(S,d)$ (where $d$ is its metric) is a smooth surface, do this mean that:

-The metric $d$ is smooth, or smoothness doesn't depend on the metric?!! (because sometimes they say "smooth" surface without mentionning the metric)

-Or may be the surface is smooth implies that the metric is smooth ? !!

Please discuss, and thanks for any answer.

Answer 1

The metric and the surface being smooth are two unrelated things: the first one means that $S$ is a topological manifold endowed with a smooth atlas, while the second says that the map $d:S\to T^*S\otimes T^*S$ is a smooth section, where the smooth structure on the fiber bundle $T^*S\otimes T^*S$ is induced by the one on $S$ (and $d$ has its values in symmetric tensors).

Note that this is an advanced definition of the smoothness of $d$, and the first one you are likely to meet is rather: for all smooth vector fields $X:S\to TS$, the map $S\to\mathbb{R}$ defined by $x\mapsto d(X_x,X_x)$ is smooth.

Answer 2

There are essentially two concepts that are not related.

First, we say that a topological space $S$ is a smooth surface, if there is an open cover $\{U_\alpha\} _{\alpha \in I}$ and for each $\alpha$, a homeomorphism $$\varphi_\alpha : U_\alpha \to \varphi_\alpha (U_\alpha) \subset \mathbb R^2$$ so that whenever $U_\alpha \cap U_\beta$ is non-empty, the composition $$ \varphi_\alpha \circ \varphi^{-1}_\beta : \varphi_\beta(U_\alpha \cap U_\beta) \to \varphi_\alpha (U_\alpha \cap U_\beta)$$ is a smooth mapping between open sets in $\mathbb R^2$.

We also require that $S$ is Hausdarff and paracompact. The point is, in the definition of a smooth surface no metric is involved.

In the study of Riemannian geometry, one gives at each tangent plane $T_xM$ an inner product $g(x)$ (which smoothly varies with $x$), so that when one has a smooth curve $\gamma : [a,b]\to S$, one can define the length of the curve: $$L(\gamma) = \int_a^b |\gamma'(t)| dt = \int_a^b \sqrt{g(\gamma'(t), \gamma'(t))} dt$$ and with this, one can define a metric $d$ on $S$, given by: for any $p,q \in S$, let $$ d(p, q) = \inf \left\{ L(\gamma) \big| \ \ \gamma: [a, b]\to S \text{ is piecewise smooth,} \ \gamma(a) = p, \gamma(q) = b\right\}.$$ One can check that

• $d$ is a metric,

• the metric topology $(S, d)$ is homeomorphic to the original topology of $S$.

So the metric $d$, as a function $S\times S \to [0,\infty)$, is always continuous. But $d$ is never a smooth function, even when the surface and the inner product are smooth. To see an example, consider the usual metric on $S = \mathbb R^2$: $$d(p, q) = \sqrt{ (p_1 - q_1)^2 + (p_2-q_2)^2}$$ note that $d$ is not differentiable whenver $p=q$. In general, $d$ is smooth at most of the points away from the diagonal, but might still fails to be differentiable at some $p\neq q$.