What is a manifold?

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• Hacker News
• I always found interesting that the English mathematical terminology has two different names for "stuff that locally looks like R^n" (manifold) and "stuff that is the zero locus of a polynomial" (variety). Other languages use the same word for both, adding maybe an adjective to specify which one is meant if not clear from the context. In Italian for example they're both "varietà"
  by genoveffo
• Lobachevsky... "the analytic and algebraic topology of locally Euclidean metrizations of infinitely differentiable Riemannian manifolds"
  by stelliosk
• Does the way "manifold" is used when describing subsets of the representational space of neural networks (e.g. "data lies on a low-dimensional manifold within the high-dimensional representation space") actually correspond to this formal definition, or is it just co-opting the name to mean something simpler (just an embedded sub-space)?

  If it is the formal definition being used, then why? Do people actually reason about data manifolds using "atlases" and "charts" of locally euclidean parts of the manifold?

  by HarHarVeryFunny
• I was reading a book on string theory and I remember the Calabi–Yau manifold

  https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold

  I'm not going to pretend to understand it all but they do make pretty pictures!

  https://www.google.com/search?q=calabi+yau+manifold+images

  by lizknope
• A manifold is a surface that you can put a cd shaped object on in any place on the surface, you can change the radius of the cd but it has to have some radius above 0.
  by ChrisGreenHeur
• This reminds me of how physicists will define a tensor. So a second rank tensor is the object that transforms according as second rank tensor when the basis (or coordinates) changes. You might find it circular reasoning but it is not, This transformation property is what distinguishes tensors (of any rank) from mere arrays of numbers.

  Looking at things from abstract view does allow us not to worry about how we visualize the geometry which is actually hard and sometimes counter intuitive.

  by elashri
• This is a very informative article about the history of manifolds and their significance. Don’t let the title fool you into this being just a definition.

  It’s actually much more well written than the majority or articles we usually come across.

  by tetrisgm
• I first learned about manifolds through Introduction to Smooth Manifolds by John M. Lee. The book is dense but beautifully structured, guiding you from basic topology to smooth maps and tangent spaces with clear logic. It demands focus, yet every definition builds toward a deeper picture of how geometry works beneath the surface. Highly recommended.
  by tamnd