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The Wikipedia's article for geometry is somehow overwhelming. To make things clear, allow me to ask some questions:

  1. I wonder if "geometry" can be defined as the study of a metric space (possibly with or without other structures)?

  2. Any thing more general than metric space (such as uniform spaces and topological spaces) is not in the scope of "geometry"?

  3. Does "geometry" assume the set under study to have some algebraic structure?

    Also there is algebraic geometries.

  4. Is the underlying set a topological vector space, normed space, inner product space, or even Euclidean space?

  5. Since projective and affine spaces are pure algebraic concepts without metrics, why are there "projective geometry" and "affine geometry"?

Thanks and regards!

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    Differential geometry doesn't generally assume a metric. Even if we insist that a differential structure alone is not sufficient to merit the label geometry, once we include a connection, we have geodesics, and I expect you will want to call it geometry. But there need not be a metric.2012-03-17
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    @yasmar: Thanks! (1) So are you saying it is connection that merit the label geometry? (2) Can this view be unified with geometries on inner product space, on (pure algebraic) projective and affine spaces?2012-03-17
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    Hi @Tim, I would certainly not go so far as to say that a connection is what is needed for geometry. In fact, I think the question in the title of your post is a good one, but I doubt there is any universally accepted answer. I would like to see more opinions expressed. I think 'geometry' can happen even in the absence of a metric, and a connection is an example, but there must be other examples of structures that don't imply a connection, or even a manifold. I can't competently answer your second question in your comment. I hope somebody else will.2012-03-17

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