Show that there is no immersion of S^n into R^n

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SUMMARY

The discussion focuses on proving that there is no immersion of the n-dimensional sphere S^n into n-dimensional Euclidean space R^n. An immersion is defined as a smooth function f: S^n → R^n where the differential D_p f maps the tangent space T_p S^n to the tangent space T_{f(p)} R^n. The conclusion is that such an immersion cannot exist for n > 0 due to topological constraints, specifically the invariance of domain theorem and the properties of manifolds.

PREREQUISITES
  • Understanding of differential geometry concepts, particularly immersions and tangent spaces.
  • Familiarity with the properties of manifolds and the invariance of domain theorem.
  • Knowledge of smooth functions and their differentiability.
  • Basic understanding of topology, specifically the characteristics of spheres and Euclidean spaces.
NEXT STEPS
  • Study the invariance of domain theorem in topology.
  • Explore the properties of smooth manifolds and their embeddings.
  • Learn about differential forms and their applications in geometry.
  • Investigate the implications of immersion and embedding in higher-dimensional topology.
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Mathematicians, particularly those specializing in topology and differential geometry, as well as students studying advanced concepts in manifold theory.

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show that there is no immersion of [/n] into [R][/n]
 
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show that there is no immersion of S^n into R^n
 


By definition;

f : S^{n}\rightarrow R^n is an immersion if;

D_{p}f : T_{p}S^n \rightarrow T_{f(p)}R^n
 

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