Orientation reversing diffeomorphism but no isometry?

Question

Is it possible that an oriented Riemannian manifold $(M,g)$ with a large isometry group $\text{Isom}(M,g)$ has an orientation reversing self-diffeomorphism but no orientation reversing self-isometry, i.e. $\text{Isom}(M,g) \subseteq \text{Diff}^+(M)$? "Large" should mean something like $\dim \text{Isom}(M,g) \geq 1$.

Can one detect such manifolds resp. Riemannian metrics in an easy way, if they exist?

Answer 1

This is probably not what you want, but you can take any manifold that admits an orientation-reversing self diffeomorphism and equip it with a generic metric that does not admit any self-isometry to get an example. So in a way, this what should happen "typically".

Answer 2

I believe if you take a connected sum of countably many copies of a rigid Riemannian manifold in a careful way, you should get a (noncompact) Riemannian manifold with isometry group $\mathbb Z$, all orientation preserving.

E.g. take a rigid sphere embedded in $\mathbb R^3$, pick antipodal pairs of $\epsilon$ discs in the sphere carefully so that rigidity is preserved after removing both, and stack $\mathbb Z$ copies of it on top of each other, then smooth out the connections in a consistent way. Then you have a topological open cylinder with isometry group $\mathbb Z$.

Edit:

Moreover, for any group $G$ and set of generators $X \subset G$, the directed Cayley graph of $G$, with edges labeled by elements of $X$, has automorphism group $G$. If the graph is "small enough", one should be able to fatten this edge-colored directed graph into a Riemannian 2-manifold, in such a way that the isometries of the Riemannian manifold are precisely the isometries induced by graph automorphisms, in particular all isometries being orientation preserving.

To achieve this, one would take a family of Riemannian 2-manifolds with boundary, namely a punctured sphere $V$ (or, to accommodate countably infinitely many punctures, a punctured infinite-length cylinder) with the connected components $\partial V$ in one-to-one correspondence with $X \times \{0, 1\}$, and a family of topological closed cylinders $C_x$ for $x \in X$, with the two connected components of $\partial C_x$ in explicit one-to-one correspondence with $\{0, 1\}$. The idea here is that $V$ takes place of a vertex, and $C_x$ takes place of the respective edges.

Construct these pieces in a way to conspire that

1. $\operatorname{Isom}(C_x, C_{x'})$ is empty when $x \neq x'$ and trivial when $x = x'$, and similarly $V$ should be a distinct isometry type from the rest,
2. For each of the components $S$ of $\partial C_x$, there is a unique component $S'$ of $\partial V$ and a unique gluing map $g: S \to S'$ such that $(V \sqcup C_x)/g$ is automatically a Riemannian manifold (i.e., the Riemannian metrics are compatible at the boundary). Imagine each boundary having a distinctive shape, and only complementary shapes can be glued together. If $S$ is the component of $\partial C_x$ associated with $j \in \{0, 1\}$, then $S'$ is the component of $\partial V$ associated with $(x, j)$.

These criteria are not quite sufficient to ensure that gluing pieces together into a graph will have only isometries induced by graph automorphisms - because the data of the boundaries are lost in gluing. In order to make the boundaries distinctive, I propose an additional technical criterion. Start with fixing an $\epsilon>0$. A non-contractible loop $\gamma: S^1 \to M$ of circumference $< \epsilon$ where $M = V$ or $M = C_x$ for some $x \in X$, is going to be called $\epsilon$-homotopic to $\gamma': S^1 \to M$ if there is a homotopy $\gamma \to \gamma'$ through loops of circumference $< \epsilon$.

3. Every non-contractable loop in $M=V$ or $M=C_x$ of circumference $< \epsilon$ is $\epsilon$-homotopic to a unique connected component of $\partial M$.
4. The union of one $\epsilon$-homotopy class associated to a connected component $S$ of $\partial M$ is an open subset $S \subset O_S \subset M$, with compact closure $\overline {O_S}$, of distance $>\epsilon$ from all other such closures of unions of $\epsilon$-homotopy classes $\overline{O_{S'}}$.
5. Deleting all $O_S$ from $M$ yields a new Riemannian manifold $M'$ with boundary, and all isometry requirements we placed on $M$ should also apply to $M'$, i.e., $\operatorname{Isom}(C_x', C_{x'}') = \emptyset$ if $x \neq x'$ and trivial if $x = x'$, and $V'$ is its own distinct rigid Riemannian 2-manifold with boundary. Note that all loops of circumference $< \epsilon$ in $M'$ are therefore contractible.

This should ensure that the gluing boundaries are marked by non-contractible loops of circumference $<\epsilon$.

With the above construction in mind, I contend that the following are all true, in order of decreasing plausibility:

1. Every finite group is the isometry group of a topological (compact w/o boundary) torus.
2. Every finitely generated group is the isometry group of a non-compact Riemannian 2-manifold.
3. Every countably infinite group is the isometry group of a non-compact Riemannian 2-manifold.

To achieve isometry groups which are continuous is a whole other challenge, which I can't say I'm optimistic is possible within the constraints of OP.

Credit to Mike Miller Eismeier for helping me figure out this should be possible.