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A totally geodesic map between two Riemannian manifolds is a map that carries geodesics of the domain manifold to geodesics of the target/co-domain manifold, i.e. following the definition from the introduction section of this paper $F:M\to N$ is totally geodesic if for every geodesic $\gamma\subset M, F(\gamma)\equiv F \circ \gamma \subset N$ is a geodesic in $N.$ This is fairly intuitive and one can see that every local self isometry of $M$ is totally geodesic. This is also a generalization of linear maps between two vector spaces if I'm correct.

However, I'd like to get some simple examples and if possible, characterizations of these maps. To start with, is there any characterizations or way to construct examples of totally geodesic maps from $M$ to $\mathbb{R}?$ Clearly, constant maps are there but I want some non trivial examples. We can start with simple questions like:

What are all the totally geodesic maps from:

  1. $\mathbb{R}^d \to \mathbb{R}?$ The answer to the question when these maps fix the origin is : all linear projections.

  2. $\mathbb{S}^d \to \mathbb{R}?$ One could be deceived into thinking it's the stereographic projection followed by a linear projection, but I think it's not, since the stereographic projection has domain $\mathbb{S}^d$ minus a point, and this point maps to infinity. It seems to me that there's no such maps because if it's continuous, it's image must be a compact subset of the real line, and that's not a geodesic.

  3. Generalizing 2), it seems to me that the all totally geodesic maps from a compact $M$ to $\mathbb{R}$ must be constant and hence, trivial.

  4. Given 2) and 3), one can now ask: what about totally geodesic maps from a noncompact Riemannian manifold to $\mathbb{R}?$ I guess one can start with the simply connected non-positive curvature manifolds, that are diffeomorphic to $\mathbb{R}^d$ by their exponential map (Cartan-Hadamard theorem). Maybe this is a good start, so in this case, is any totally geodesic map the combination of inverse exponential and linear projection?

  5. Some relevant literature with lots of examples would be appreciated!

Learning Math
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  • About (4). Pick a preimage of $0$ in $\mathbb{R}$ call it $m$. Then measure distance from $m$ to any given point $x$ by the unique minimising geodesic. This gives you the map $M \to \mathbb{R}$ up to sign which can be patched up by continuity. – Radost Aug 03 '23 at 20:47
  • @Radost Thank you for your comment. I'd appreciate if you clarify it a bit further? First, why is the minimizing geodesic unique - are you assuming that $Cut(M)=\emptyset,$ or $x\in Cut(m)?$ I also couldn't understand how $\pm d(x,m)$ gives us the totally geodesic map from $M\to R$? Could you elaborate this part, maybe as an answer? – Learning Math Aug 03 '23 at 21:47
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    https://math.stackexchange.com/questions/4746855/totally-geodesic-diffeomorphism-and-isometry – Deane Aug 04 '23 at 04:01
  • @Deane Thanks, the answer to this other question in the above link is no, since the manifolds don't even need to be of same dimension in order for us to talk about any global/local isometries. – Learning Math Aug 04 '23 at 15:42

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