Skip to main content
MathOverflow

Return to Answer

added 2 characters in body
Source Link
Vít Tuček
Vít Tuček
  • 8.6k
  • 2
  • 30
  • 61

Apparently, Hopf proved in [H] that the projective space $\mathbb{R}P^3$ embedds into $\mathbb{R}^5$ and Hantzsche [Ha] proved that it cannot be embedded into $\mathbb{R}^4$. Since this projective space is isomorphic to $\mathrm{SO}(3)$ as was noted by Paul Siegel, this at least gives a lower bound. Let me just add that $\mathrm{Spin}(3)$ is (isomorphic to) the unit sphere in quaternions and hence this double cover of $\mathrm{SO}(3)$ is embeddable into $\mathbb{R}^4$.

[H] H.Hopf, "Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Raume", Vierteljschr Naturforsch Gesellschaft Zurich 85 (1940) 165-177. [Ha]

[Ha] W.Hantzsche, "Einlagerung von Mannigfaltigkeiten in euklidische Raume", Math Zeit 43 (1938) 38-58.

Apparently, Hopf proved in [H] that the projective space $\mathbb{R}P^3$ embedds into $\mathbb{R}^5$ and Hantzsche [Ha] proved that it cannot be embedded into $\mathbb{R}^4$. Since this projective space is isomorphic to $\mathrm{SO}(3)$ as was noted by Paul Siegel, this at least gives a lower bound. Let me just add that $\mathrm{Spin}(3)$ is (isomorphic to) the unit sphere in quaternions and hence this double cover of $\mathrm{SO}(3)$ is embeddable into $\mathbb{R}^4$.

[H] H.Hopf, "Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Raume", Vierteljschr Naturforsch Gesellschaft Zurich 85 (1940) 165-177. [Ha] W.Hantzsche, "Einlagerung von Mannigfaltigkeiten in euklidische Raume", Math Zeit 43 (1938) 38-58.

Apparently, Hopf proved in [H] that the projective space $\mathbb{R}P^3$ embedds into $\mathbb{R}^5$ and Hantzsche [Ha] proved that it cannot be embedded into $\mathbb{R}^4$. Since this projective space is isomorphic to $\mathrm{SO}(3)$ as was noted by Paul Siegel, this at least gives a lower bound. Let me just add that $\mathrm{Spin}(3)$ is (isomorphic to) the unit sphere in quaternions and hence this double cover of $\mathrm{SO}(3)$ is embeddable into $\mathbb{R}^4$.

[H] H.Hopf, "Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Raume", Vierteljschr Naturforsch Gesellschaft Zurich 85 (1940) 165-177.

[Ha] W.Hantzsche, "Einlagerung von Mannigfaltigkeiten in euklidische Raume", Math Zeit 43 (1938) 38-58.

added 189 characters in body
Source Link
Vít Tuček
Vít Tuček
  • 8.6k
  • 2
  • 30
  • 61

Apparently, Hopf proved in [H] that the projective space $\mathbb{R}P^3$ embedds into $\mathbb{R}^5$ and Hantzsche [Ha] proved that it cannot be embedded into $\mathbb{R}^4$. Since Since this porjectiveprojective space is isomorphic to $\mathrm{SO}(3)$ as was noted by Paul Siegel, this should answer the questionat least gives a lower bound. Let me just add that $\mathrm{Spin}(3)$ is (isomorphic to) the unit sphere in quaternions and hence this double cover of $\mathrm{SO}(3)$ is embeddable into $\mathbb{R}^4$.

[H] H.Hopf, "Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Raume", Vierteljschr Naturforsch Gesellschaft Zurich 85 (1940) 165-177. [Ha] W.Hantzsche, "Einlagerung von Mannigfaltigkeiten in euklidische Raume", Math Zeit 43 (1938) 38-58.

Apparently, Hopf proved in [H] that the projective space $\mathbb{R}P^3$ embedds into $\mathbb{R}^5$. Since this porjective space is isomorphic to $\mathrm{SO}(3)$ as was noted by Paul Siegel, this should answer the question. Let me just add that $\mathrm{Spin}(3)$ is (isomorphic to) the unit sphere in quaternions and hence this double cover of $\mathrm{SO}(3)$ is embeddable into $\mathbb{R}^4$.

[H] H.Hopf, "Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Raume", Vierteljschr Naturforsch Gesellschaft Zurich 85 (1940) 165-177.

Apparently, Hopf proved in [H] that the projective space $\mathbb{R}P^3$ embedds into $\mathbb{R}^5$ and Hantzsche [Ha] proved that it cannot be embedded into $\mathbb{R}^4$. Since this projective space is isomorphic to $\mathrm{SO}(3)$ as was noted by Paul Siegel, this at least gives a lower bound. Let me just add that $\mathrm{Spin}(3)$ is (isomorphic to) the unit sphere in quaternions and hence this double cover of $\mathrm{SO}(3)$ is embeddable into $\mathbb{R}^4$.

[H] H.Hopf, "Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Raume", Vierteljschr Naturforsch Gesellschaft Zurich 85 (1940) 165-177. [Ha] W.Hantzsche, "Einlagerung von Mannigfaltigkeiten in euklidische Raume", Math Zeit 43 (1938) 38-58.

Source Link
Vít Tuček
Vít Tuček
  • 8.6k
  • 2
  • 30
  • 61

Apparently, Hopf proved in [H] that the projective space $\mathbb{R}P^3$ embedds into $\mathbb{R}^5$. Since this porjective space is isomorphic to $\mathrm{SO}(3)$ as was noted by Paul Siegel, this should answer the question. Let me just add that $\mathrm{Spin}(3)$ is (isomorphic to) the unit sphere in quaternions and hence this double cover of $\mathrm{SO}(3)$ is embeddable into $\mathbb{R}^4$.

[H] H.Hopf, "Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Raume", Vierteljschr Naturforsch Gesellschaft Zurich 85 (1940) 165-177.