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Brian Lins
Brian Lins
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If a real analytic variety $V$ in $\mathbb{R}^n$ is both bounded and contractible, is it true that $V$ must be a single point?

Here a real analytic variety is the set of zeros of a real analytic function $f: \mathbb{R}^n \rightarrow \mathbb{R}$.

This is certainly true if $V$ is a compact real analytic manifold (without boundary). But what about varieties that are not manifolds? Answers in other settings (complex analytic varieties or real algebraic varieties) would be interesting too.

If a real analytic variety $V$ in $\mathbb{R}^n$ is both bounded and contractible, is it true that $V$ must be a single point?

This is certainly true if $V$ is a compact real analytic manifold (without boundary). But what about varieties that are not manifolds? Answers in other settings (complex analytic varieties or real algebraic varieties) would be interesting too.

If a real analytic variety $V$ in $\mathbb{R}^n$ is both bounded and contractible, is it true that $V$ must be a single point?

Here a real analytic variety is the set of zeros of a real analytic function $f: \mathbb{R}^n \rightarrow \mathbb{R}$.

This is certainly true if $V$ is a compact real analytic manifold (without boundary). But what about varieties that are not manifolds? Answers in other settings (complex analytic varieties or real algebraic varieties) would be interesting too.

Source Link
Brian Lins
Brian Lins
  • 630
  • 3
  • 9

Contractible real analytic varieties

If a real analytic variety $V$ in $\mathbb{R}^n$ is both bounded and contractible, is it true that $V$ must be a single point?

This is certainly true if $V$ is a compact real analytic manifold (without boundary). But what about varieties that are not manifolds? Answers in other settings (complex analytic varieties or real algebraic varieties) would be interesting too.