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Here, I try to give a exemple for each of the number one and three curves with a convex polyhedron where I just draw the net. The geodesic are the dots lines (which are staight on the net). I think that one can also construct the second one as well by changing the angles of the triangle in the first exemple.

I don't know any theorem that gives a general answer for your question. My only gess is that if you don't have a trivial estimate that says that on a subset $B$, $\int_B K \geq 4\pi $, then we should be able to construct a exemple. [![enter image description here][1]][1] [![enter image description here][3]][3]

EDIT : I just give some details of what anton says in his comment : in each small loop the integral of curvature is $\pi+\alpha_i$ with $\alpha_i$ the angle of the crossing. Because the curvature is positive the sum of the angles of the triangle is larger than $\pi$ and then the total curvature is larger than $3\pi+ \sum \alpha_i > 4\pi$. (However with a polyhedron we can still obtain an equality) [1]: https://i.sstatic.net/LNoH4.png [2]: https://i.sstatic.net/otkxy.png [3]: https://i.sstatic.net/3prhR.png

Here, I try to give a exemple for each of the number one and three curves with a convex polyhedron where I just draw the net. The geodesic are the dots lines (which are staight on the net). I think that one can also construct the second one as well by changing the angles of the triangle in the first exemple.

I don't know any theorem that gives a general answer for your question. My only gess is that if you don't have a trivial estimate that says that on a subset $B$, $\int_B K \geq 4\pi $, then we should be able to construct a exemple.

EDIT : I just give some details of what anton says in his comment : in each small loop the integral of curvature is $\pi+\alpha_i$ with $\alpha_i$ the angle of the crossing. Because the curvature is positive the sum of the angles of the triangle is larger than $\pi$ and then the total curvature is larger than $3\pi+ \sum \alpha_i > 4\pi$. (However with a polyhedron we can still obtain an equality)

Here, I try to give a exemple for each of the number one and three curves with a convex polyhedron where I just draw the net. The geodesic are the dots lines (which are staight on the net). I think that one can also construct the second one as well by changing the angles of the triangle in the first exemple.

I don't know any theorem that gives a general answer for your question. My only gess is that if you don't have a trivial estimate that says that on a subset $B$, $\int_B K \geq 4\pi $, then we should be able to construct a exemple.

Here, I try to give a exemple for each of the number one and three curves with a convex polyhedron where I just draw the net. The geodesic are the dots lines (which are staight on the net). I think that one can also construct the second one as well by changing the angles of the triangle in the first exemple.

I don't know any theorem that gives a general answer for your question. My only gess is that if you don't have a trivial estimate that says that on a subset $B$, $\int_B K \geq 4\pi $, then we should be able to construct a exemple. [![enter image description here][1]][1] [![enter image description here][3]][3]

EDIT : I just give some details of what anton says in his comment : in each small loop the integral of curvature is $\pi+\alpha_i$ with $\alpha_i$ the angle of the crossing. Because the curvature is positive the sum of the angles of the triangle is larger than $\pi$ and then the total curvature is larger than $3\pi+ \sum \alpha_i > 4\pi$. (However with a polyhedron we can still obtain an equality) [1]: https://i.sstatic.net/LNoH4.png [2]: https://i.sstatic.net/otkxy.png [3]: https://i.sstatic.net/3prhR.png

Here, I try to give a exemple for each of the three curves with a convex polyhedron where I just draw the net. The geodesic are the dots lines (which are staight on the net). But I don't know any theorem that gives a general answer for your question. My only gess is that if you don't have a trivial estimate that says that on a subset $B$, $\int_B K \geq 4\pi $, then we should be able to construct a exemple.

Here, I try to give a exemple for each of the number one and three curves with a convex polyhedron where I just draw the net. The geodesic are the dots lines (which are staight on the net). I think that one can also construct the second one as well by changing the angles of the triangle in the first exemple.

I don't know any theorem that gives a general answer for your question. My only gess is that if you don't have a trivial estimate that says that on a subset $B$, $\int_B K \geq 4\pi $, then we should be able to construct a exemple.