Definition of a circle on a manifold

Question

I'm reading Tristan Needham's Visual Differential Geometry and Forms, specifically the start of the book where he's giving a rough intuitive idea of what non-Euclidean geometry is.

He gives the example of a crookneck squash:

The analogy for a geodesic curve between, say $c$ and $d$ is a string stretched taut between the two points (imagine the string on the inside of the surface), with the disclaimer that it is between two sufficiently close points that we can find a unique length-minimizing geodesic segment.

The string analogy makes sense. Now on to defining a circle on such a surface:

(the figure) shows how we may then define, for example, a “circle of radius $r$ and centre $c$” as the locus of points at distance $r$ from $c$. To construct this geodesic circle we may take a piece of string of length $r$, hold one end fixed at $c$, then (keeping the string taut) drag the other end round on the surface.

Maybe I'm nitpicking, but how do we know that the piece of string is length $r$ anyways? Seems like the above definition assumes that we've already measured the string in the ambient space and then use it to construct a circle on the surface.

The book states that there's intrinsic and extrinsic geometry. How would we construct a circle in intrinsic geometry? Since in that case, we only have the surface to work with and nothing else - no ambient space.

Answer 1

Suppose the surface $M$ comes equipped with a Riemannian metric $g$, which is the same thing as smoothly varying choice of inner product on $M$. That is, $g$ assigns to each $p$ an inner product $g_p$ on $T_pM$. Then you can define the distance between two points to be: $d(p,q)=\inf\{\ell(\gamma):\gamma(0)=p\:\text{and}\:\gamma(1)=q\}.$ Here, $\gamma:[0,1]\to M$ is a smooth path connecting $p$ to $q$ and you can define $$ \ell(\gamma)=\int_0^1\lVert \gamma'(t)\rVert dt. $$ The expression $\lVert \gamma'(t)\rVert$ means $\sqrt{g_{\gamma(t)}( \gamma'(t),\gamma'(t))}$ Once you have this definition, you could try to define the circle of radius $r$ about a point to be $\{q\in M:d(p,q)=r\}$. This is a valid definition, but the shape of this "circle" might actually be contrary to intuition when $r$ is large. Luckily, there is a result which often goes by the name of Gauss's Lemma that says (roughly) that there are neighborhoods $0\in U\subseteq T_pM$ and $V\ni p$ so that the "exponential map" $\exp:T_pM \to V$ restricts to a diffeomorphism carrying circles of small radius $r$ about $0$ in $U$ onto $\{q\in M:d(p,q)=r\}$.

The exponential map can be thought of as taking a point $p\in M$ along with $v\in T_pM$, then stretching a length of string from $p$ to $v$ in $T_pM$ then pulling it taught along the surface so that it forms a "geodesic" emanating from $p$.