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In the article

Frankel, T., Manifolds with positive curvature, Pac. J. Math. 11, 165-174 (1961). ZBL0107.39002., there is a theorem (Theorem 1), says that in any complete connected manifold with positive sectional curvature any two compact totally geodesic submanifolds such that the sum of their dimensions is greater or equal to that of the original manifold, must have non-empty intersection.
But if I take any two disjoint geodesic segments in $\mathbb{S}^2$, then these two segments are compact and totally geodesic submanifolds but they are disjoint. I am very confused about that. I can not understand where I am doing wrong.
Please help.

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1 Answer 1

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Geodesic segments in $S^2$ are great circles. How are you taking disjoint great circles in $S^2$? Recall that great circles are intersections of a plane through the origin in $\mathbb{R}^3$ with $S^2$. Any two planes in $\mathbb{R}^3$ intersect non-trivially in at least a line by dimensional reasons, and this intersection gives (at least) two points of intersection on the sphere $S^2$ (the intersection of a line with $S^2$).

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  • $\begingroup$ But I am taking only a small segment of a geodesic, e.g, two disjoint intervals of the equator. $\endgroup$
    – MAS
    Commented Apr 26, 2018 at 19:18
  • $\begingroup$ @chandanmondal Then these are not submanifolds, if you are taking "closed" segments, or not compact, if you are taking "open" ones. $\endgroup$
    – Aloizio Macedo
    Commented Apr 26, 2018 at 19:35
  • $\begingroup$ Ok I got it. Its a submanifold with boundary. $\endgroup$
    – MAS
    Commented Apr 26, 2018 at 19:44

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