Homotopy Groups, Focal Points and Totally Geodesic Immersions

Details Homotopy Groups, Focal Points and Totally Geodesic Immersions Homotopy Groups, Focal Points and Totally Geodesic Immersions

2011-08-08

arxiv.org/abs/1108.1822v1

Mathematics

Differential Geometry

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Scientific paper

Let $f:\Si^{m-1}\to M^m$ be a totally geodesic immersion of a closed manifold $\Si$ in a complete Riemannian manifold $M$ and $g:N^n\to M^m$ an isometric immersion without focal points of a complete manifold $N$. If $\Si$ has finite fundamental group then $N$ is compact with finite fundamental group and we have: {enumerate}[(1)] if $m-n=1$ then $\Si$ and $N$ have the same universal covering, and the homomorphism $f_*^i:\pi_i(\Si)\to \pi_i(M)$ is a monomorphism for $i=1$ and an isomorphism for $i\ge 2$; if $m-n\ge 2$ then it holds that: {enumerate}[(a)] $f(\Si)\cap g(N)=\emptyset$; $M$ is noncompact with finite fundamental group; the homomorphism $\iota_*^i:\pi_i(f(\Si))\to \pi_i(M)$, induced by the inclusion $\iota:f(\Si)\to M$, is surjective for $1\le i\le m-n-1$; if $m-n=2$ then $f_*^i:\pi_i(\Si)\to \pi_i(M)$ is a monomorphism for $i=2$ and an isomorphism for $i\ge 3$; {enumerate} if $m-n\geq 3$ then the following statements hold: {enumerate}[(a)] the map $f_*^i:\pi_i(\Si)\to \pi_i(M)$ is a monomorphism for $i=1$, an isomorphism for $2\le i\le m-n-2$ and an epimorphism for $i=m-n-1$; if $f$ is an embedding then $f_*^1:\pi_1(\Si)\to \pi_1(M)$ is surjective and this, together with $\ref{it6}$ above, implies that $f$ is $(m-n-1)$-connected. {enumerate} {enumerate}

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