Skip to main content
MathOverflow

Return to Answer

replaced http://mathoverflow.net/ with https://mathoverflow.net/
Source Link
Community Bot
Community Bot
  • 1
  • 2
  • 3

This is an alternative way to see this question. By argument given by the current answercurrent answer given by Ehud Meir we may assume that $R_1$ is the identity matrix. Then the hypotheses imply that for $i=2,\cdots,d$ the maps $v\rightarrow R_i v$ are $d-1$ independent vector fields on the unit sphere $S^{d-1}$. But it is known that the maximal number of independent vector fields on the unit sphere $S^{n-1}$ is $n-1$ if and only if $n=1, 2, 4 $ or $8$. This follows from a much stronger result due to J. F. Adams (Annals of Mathematics, 1962)

This is an alternative way to see this question. By argument given by the current answer given by Ehud Meir we may assume that $R_1$ is the identity matrix. Then the hypotheses imply that for $i=2,\cdots,d$ the maps $v\rightarrow R_i v$ are $d-1$ independent vector fields on the unit sphere $S^{d-1}$. But it is known that the maximal number of independent vector fields on the unit sphere $S^{n-1}$ is $n-1$ if and only if $n=1, 2, 4 $ or $8$. This follows from a much stronger result due to J. F. Adams (Annals of Mathematics, 1962)

This is an alternative way to see this question. By argument given by the current answer given by Ehud Meir we may assume that $R_1$ is the identity matrix. Then the hypotheses imply that for $i=2,\cdots,d$ the maps $v\rightarrow R_i v$ are $d-1$ independent vector fields on the unit sphere $S^{d-1}$. But it is known that the maximal number of independent vector fields on the unit sphere $S^{n-1}$ is $n-1$ if and only if $n=1, 2, 4 $ or $8$. This follows from a much stronger result due to J. F. Adams (Annals of Mathematics, 1962)

deleted 1 character in body
Source Link
Name
Name
  • 2k
  • 14
  • 21

This is an alternative way to see this question. By argument given by the current answer given by Ehud Meir we may assume that $R_1$ is the identity matrix. Then the hypotheses imply that for $i=2,\cdots,d-1$$i=2,\cdots,d$ the mapmaps $v\rightarrow R_i v$ are $d-1$ independent vector fields on the unit sphere $S^{d-1}$. But it is known that the maximal number of independent vector fields on the unit sphere $S^{n-1}$ is $n-1$ if and only if $n=1, 2, 4 $ or $8$. This follows from a much stronger result due to J. F. Adams (Annals of Mathematics, 1962)

This is an alternative way to see this question. By argument given by the current answer given by Ehud Meir we may assume that $R_1$ is the identity matrix. Then the hypotheses imply that for $i=2,\cdots,d-1$ the map $v\rightarrow R_i v$ are $d-1$ independent vector fields on the unit sphere $S^{d-1}$. But it is known that the maximal number of independent vector fields on the unit sphere $S^{n-1}$ is $n-1$ if and only if $n=1, 2, 4 $ or $8$. This follows from a much stronger result due to J. F. Adams (Annals of Mathematics, 1962)

This is an alternative way to see this question. By argument given by the current answer given by Ehud Meir we may assume that $R_1$ is the identity matrix. Then the hypotheses imply that for $i=2,\cdots,d$ the maps $v\rightarrow R_i v$ are $d-1$ independent vector fields on the unit sphere $S^{d-1}$. But it is known that the maximal number of independent vector fields on the unit sphere $S^{n-1}$ is $n-1$ if and only if $n=1, 2, 4 $ or $8$. This follows from a much stronger result due to J. F. Adams (Annals of Mathematics, 1962)

Source Link
Name
Name
  • 2k
  • 14
  • 21

This is an alternative way to see this question. By argument given by the current answer given by Ehud Meir we may assume that $R_1$ is the identity matrix. Then the hypotheses imply that for $i=2,\cdots,d-1$ the map $v\rightarrow R_i v$ are $d-1$ independent vector fields on the unit sphere $S^{d-1}$. But it is known that the maximal number of independent vector fields on the unit sphere $S^{n-1}$ is $n-1$ if and only if $n=1, 2, 4 $ or $8$. This follows from a much stronger result due to J. F. Adams (Annals of Mathematics, 1962)