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aglearner
aglearner
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It is well known that compact manifolds of negative sectional curvature don't admit self-maps of degree $>1$. At the same time positively curved manifolds such as $S^n$ and $\mathbb CP^n$ clearly admit self-maps of degree $>1$. Moreover I guess if we take a compact Lie group and consider map $x\to x^2$ the degree of such map is $>1$ (in fact I did not check it).

Question. Let $G$ be a compact Lie group and $H$ its proper connected Lie subgroup. Is it true that $G/H$ admits self-maps of degree $>1$? If not, is there some classification of such quotients and that admit / don't admit such a self-map with $deg>1$?

It is well known that compact manifolds of negative sectional curvature don't admit self-maps of degree $>1$. At the same time positively curved manifolds such as $S^n$ and $\mathbb CP^n$ clearly admit self-maps of degree $>1$. Moreover I guess if we take a compact Lie group and consider map $x\to x^2$ the degree of such map is $>1$ (in fact I did not check it).

Question. Let $G$ be a compact Lie group and $H$ its proper connected Lie subgroup. Is it true that $G/H$ admits self-maps of degree $>1$? If not, is some classification of such quotients and that admit / don't admit such a self-map?

It is well known that compact manifolds of negative sectional curvature don't admit self-maps of degree $>1$. At the same time positively curved manifolds such as $S^n$ and $\mathbb CP^n$ clearly admit self-maps of degree $>1$. Moreover I guess if we take a compact Lie group and consider map $x\to x^2$ the degree of such map is $>1$ (in fact I did not check it).

Question. Let $G$ be a compact Lie group and $H$ its proper connected Lie subgroup. Is it true that $G/H$ admits self-maps of degree $>1$? If not, is there some classification of such quotients and that admit / don't admit a self-map with $deg>1$?

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aglearner
aglearner
  • 14.3k
  • 8
  • 41
  • 99

Compact homogeneous spaces that admit a self map of degree >1

It is well known that compact manifolds of negative sectional curvature don't admit self-maps of degree $>1$. At the same time positively curved manifolds such as $S^n$ and $\mathbb CP^n$ clearly admit self-maps of degree $>1$. Moreover I guess if we take a compact Lie group and consider map $x\to x^2$ the degree of such map is $>1$ (in fact I did not check it).

Question. Let $G$ be a compact Lie group and $H$ its proper connected Lie subgroup. Is it true that $G/H$ admits self-maps of degree $>1$? If not, is some classification of such quotients and that admit / don't admit such a self-map?