Riemannian manifolds with small geodesics and bounded curvature

Question

Let $(M,g)$ be a compact riemannian manifold with sectional curvature $|K_g| \leq 1$. A lemma due to Klingenberg asserts that then either the injectivity radius $i_g \geq \pi$ or $(M,g)$ contains a geodesic loop $\gamma$ of length $L(\gamma) = 2i_g$.

Let $n = 2$. Suppose for a moment that $i_g < \pi$ and $M$ is orientable. Note that $\gamma$ is automatically simple. (I am note sure how often this occurs. An example, however, is a thin long cylinder with two balloon-like 2-spheres attached.) Then $\gamma$ cuts $M$ into two pieces.

Intuition says that these two pieces should be rather large, as the manifold cannot curve together very quickly due to the curvature condition. By another Klingenberg lemma, known as the "Long Homotopy Lemma", any null-homotopy $\gamma_t$ of $\gamma_0 = \gamma$ must sweep through some curve $\gamma_{t_0}$ of length $L(\gamma_{t_0}) \geq \pi$. (This lemma holds for all $n$ and non-orientable manifolds. It is Exercise 1 in Chapter 10 of do Carmo's text book.)

To some extend, this is the kind of theorem I am looking for. It asserts that away from $\gamma$, $(M,g)$ must be slightly larger (on the scale of the curvature bound) than near $\gamma$. However, I'm looking for some result describes the geometry of $M$ away from $\gamma$ in a "more global" fashion. For instance: Is there any estimate of the diameter or the volume of the pieces that $M$ is cut into?

In view of the examples given by horse with no name, one should assume that $M$ has $g(M) \neq 0$ (or that $M$ has some other non-vashing characteristic class in the case that $n > 2$).

As pointed out in the comments and Anton Petrunin's answer, $M$ should either be a sphere or $\gamma$ must also be assumed to cut $M$ in half.

Answer 1

Cheeger's examples:

Arbitrarily small volume (but bounded diameter and $i_g\rightarrow 0$ ): Flat torus $S^1(1)\times S^1(\frac{1}{n})$ where the first circle has diameter 1 and the second diameter $1/n$.

Arbitrarily large diameter (bounded volume and $i_g\rightarrow 0$): Flat torus $S^1(n)\times S^1(\frac{1}{n})$ where the first circle has diamter $n$ and the second diamter $1/n$.

Answer 2

It seems that you want $M$ to be a sphere (although you do not mention it).

The natural example is a tube with curvature $-1$ closed by two spherical cups with curvature $1$. (This is a surface of revolution.) This example should have the minimal area and minimal diameter among the spheres with given length of closed geodesic. But at the moment I do not see a proof.

Some rough estimates will follow from the Klingenberg's result. Take the neighborhood $U$ of your closed geodesic of size near $e^{1/\ell}$ where $\ell$ is the length of the geodesic. Note that $U$ has to be a cylinder otherwise you end in singular point. It implies that diameter has to have order $e^{1/\ell}$.

Rough bounds for the area follow Rbega comment. They are not far from the area of the surface of revolution, somewhere in the range between $4{\cdot}\pi$ to $(4+2{\cdot}\sqrt{2}){\cdot}\pi$.