Riemannian Geometry by Peter Petersen

Author:Peter Petersen
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

Finally we mention a significant result that allows us to make strong conclusions about connectedness in positive curvature. The result will be enhanced in lemma 8.3.6.

Lemma 6.5.8 (The Connectedness Principle, Wilking, 2003).

Let M n be a compact n-manifold with positive sectional curvature. (a) If is a closed codimension k totally geodesic submanifold, then is -connected.

(b) If and are closed totally geodesic submanifolds of M with k 1 ≤ k 2 and k 1 + k 2 ≤ n, then is a nonempty totally geodesic submanifold and is -connected.

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