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Ali Taghavi
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(retract) embedding of $S^{n}$ in its unit tangent bundle

Is there an even integer $n$ such that $S^n$ can be embedded in its unit tangent bundle $$T^{1} S^n=\{(x,y)\in S^n \times S^n \mid x.y=0\}$$ What about if we pose the extra condition that the image of $S^n$ is a retract of the unit tangent bundle?

Ali Taghavi
  • 356
  • 8
  • 31
  • 123