Singa lemma

Singh's lemma is a key statement on the stability of closed geodesics in Riemannian manifolds with positive sectional curvature.

The lemma is a direct consequence of the formula for the second variation of the lengths of a one-parameter family of curves. It was used by John Singh . ^[one]

Wording

Let be $\gamma \colon [0;1]\to M$ ${\ displaystyle \ gamma \ colon [0; 1] \ to M}$ there is a geodesic in the Riemannian manifold $M$ ${\ displaystyle M}$ with positive sectional curvature and $V$ ${\ displaystyle V}$ parallel field of tangent vectors on $\gamma$ ${\ displaystyle \ gamma}$ . Then the variation $\gamma$ ${\ displaystyle \ gamma}$ in the direction $V$ ${\ displaystyle V}$ shortens its length.

More precisely, if

\gamma _{\tau }(t)=\exp _{\gamma (t)}(\tau \cdot V(t))

{\ displaystyle \ gamma _ {\ tau} (t) = \ exp _ {\ gamma (t)} (\ tau \ cdot V (t))}

and $L(\tau )$ ${\ displaystyle L (\ tau)}$ denotes the length of the curve $\gamma _{\tau }$ ${\ displaystyle \ gamma _ {\ tau}}$ then $L'(0)=0$ ${\ displaystyle L '(0) = 0}$ and $L''(0)<0$ ${\ displaystyle L '' (0) <0}$ .

Consequences

If a closed geodesic admitting a parallel vector field is not stable, that is, its length can be reduced by an arbitrarily small deformation. In particular,
- Even-dimensional oriented Riemannian manifolds with positive sectional curvature are simply connected .
- Odd-dimensional Riemannian manifolds with positive sectional curvature are oriented .
Singh's lemma was also used by ^[2] to prove that if $V$ ${\ displaystyle V}$ and $W$ ${\ displaystyle W}$ are closed geodesic submanifolds in the Riemannian manifold $M$ ${\ displaystyle M}$ with positive sectional curvature and $\dim V+\dim W\geq \dim M$ ${\ displaystyle \ dim V + \ dim W \ geq \ dim M}$ then $V$ ${\ displaystyle V}$ and $W$ ${\ displaystyle W}$ intersect.

Notes

↑ Synge, John Lighton (1936), " On the connectivity of spaces of positive curvature ", Quarterly Journal of Mathematics (Oxford Series) Vol. 7: 316–320 , DOI 10.1093 / qmath / os-7.1.316
↑ Frankel, Theodore. Manifolds with positive curvature (English) // Pacific J. Math .. - 1961. - Vol. 11 . - P. 165–174 .

[1] Synge, John Lighton (1936), " On the connectivity of spaces of positive curvature ", Quarterly Journal of Mathematics (Oxford Series) Vol. 7: 316–320 , DOI 10.1093 / qmath / os-7.1.316

[2] Frankel, Theodore. Manifolds with positive curvature (English) // Pacific J. Math .. - 1961. - Vol. 11 . - P. 165–174 .

Singa lemma

Wording

Consequences

Notes

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