Let $M$ be a complete finite volume Riemannian manifold and $\gamma : \mathbb{R}^{\geq 0} \to M$ a geodesic. Suppose that $\mathrm{im}(\gamma)$ is dense. Is it equidistributed in the Riemannian measure? That is, does $$ \lim_{T \to +\infty} \frac{1}{T} \int_0^T f(\gamma(t)) \, dt = \frac{1}{\mathrm{vol}(M)} \int_M f \, \mathrm{d vol} $$ for every $f \in C_0(M)$? [False in general; true for Nilmanifolds. True a.e. in negative curvature, where the geodesic flow is ergodic. ]

Let now $N \subset M$ be an (immersed) submanifold and $\gamma$ a geodesic of $M$ which is contained densely in $N$. Is the submanifold $N \subset M$ totally geodesic? [False in general, though true for some variants in constant negative curvature. But what if "totally geodesic" is weakened to "minimal"?]

**Added.** Asaf's answer nonethtless begs a follow-up question to 1:

- (Revised). If there is a dense geodesic, must there also be an equidistributed one? Could it in fact be that almost every geodesic is then equidistributed? Does a single dense geodesic imply ergodic geodesic flow? And in particular: does one dense geodesic imply almost all geodesics dense?

[A similar revision of 2 would instead involve the condition that almost every geodesic of $M$ that is tangent to $N$ at some point is contained in $N$; but then it should follow trivially (I think) that $N$ is totally geodesic. ]

*Note:* There is an analogy with the equidistribution and Manin-Mumford theorems, due to Szpiro, Ullmo, and Zhang, for torsion points in abelian varieties $A/\bar{\mathbb{Q}}$: For a sequence of torsion points which is eventually outside of every torsion translate of an abelian subvariety, the Dirac masses at the Galois orbits converge to the normalized Haar measure on $A(\mathbb{C})$ (where an embedding $\bar{\mathbb{Q}} \hookrightarrow \mathbb{C}$ has been fixed). Here, I would be tempted to think of a geodesic as corresponding to a Galois orbit of torsion points (either minimizes an energy functional -- or a canonical height); and of a totally geodesic subvariety as corresponding to a Galois orbit of a translate of an abelian subvariety by a torsion point (note that in the basic case of a flat torus, the totally geodesic submanifolds are precisely the subtori). The analogy is probably only superficial, but I thought it could be worth pointing out (if only because it led me to asking this question).

**Added later.** One more (final) question along the line of 2.

In algebraic geometry, we have the following general fact: For $L$ a nef line bundle on a projective variety $X$, if $\deg_LC =0$ for a Zariski-dense set of curves $C \subset X$, then $\deg_LX = 0$. (Nef =non-negative intersection numbers, =non-negative on every curve). For if $\deg_LX > 0$, Riemann-Roch and the almost vanishing of the higher cohomology of powers of nef line bundles imply that $L$ is big, hence a power of $L$ is effective. We may also do this in an arithmetic setting.

In the analogy of the preceding note which led me to consider totally geodesic submanifolds, I was misled by the Manin-Mumford theorem, which is specific to commutative group varieties and fails even for algebraic dynamical systems. Instead, subvarieties of minimal height ought to be analogous to minimal immersed submanifolds: the images of harmonic isometric immersions (which include totally geodesic ones as a particular case, and coincide with the geodesics in dimension one). Considering the previous paragraph, then, does the following question make any sense: *If the closure of a minimal submanifold happens to be an immersed submanifold, is this submanifold still minimal*?

In the same vein: *If we have a sequence of complex algebraic curves in $\mathbb{CP}^n$ (images of non-constant holomorphic maps from compact Riemann surfaces) whose supports converge to a compact real-analytic immersed submanifold $M \subset \mathbb{CP}^n$, must $M$ be a complex (algebraic) submanifold?*

testfunctions... since, e.g., Weyl's equidistribution criterion's proof seems to need more than mere continuity (though I may be mistaken). $\endgroup$ – paul garrett Jul 20 '14 at 19:42