Undergrad Is Euclidean Space Inherently Geometric or Just a Vector Space?

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The discussion explores the relationship between Euclidean space as a vector space and its geometric properties. It emphasizes that geometry requires a measure, typically an inner product, to define angles and distances, rather than relying on a geometric grid. The conversation also distinguishes between local geometric properties, such as curvature and angles, and global topological properties, highlighting that geometry is fundamentally about measurements while topology concerns the intrinsic nature of spaces. Additionally, it addresses the concept of groups operating on sets, noting that groups can exist independently of the sets they act upon. Ultimately, the discourse underscores the necessity of a metric for establishing geometric structures within vector spaces.
  • #31
martinbn said:
What if it s given a connection, wouldn't that count as geometry?

martinbn said:
What if it s given a connection, wouldn't that count as geometry?

Yes - in a more general sense.

- One could have a connection that is compatible with a metric but is not a Levi_Civita connection

- One could have a connection on the tangent bundle that is not compatible with any metric. In this case I do not see how distance relations can be derived - but not sure. Still one has curvature and parallel translation,

This thread though seems to assume metric relations of some kind.
 
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