Why is a figure 8 not a manifold?

An interesting point is that figure “8” is not a manifold because the crossing point does not locally resemble a line segment. These closed loop manifolds are the easiest 1D manifolds to think about but there are other weird cases too shown in Figure 2.

Is the sphere a Riemannian manifold?

The Riemann sphere is only a conformal manifold, not a Riemannian manifold.

What is a data manifold?

A manifold is an object of dimensionality d that is embedded in some higher dimensional space. Imagine a set of points on a sheet of paper. If we crinkle up the paper, the points are now in 3 dimensions. Many manifold learning algorithms seek to “uncrinkle” the sheet of paper to put the data back into 2 dimensions.

What is a Lorentzian manifold?

A Lorentzian manifold is an important special case of a pseudo-Riemannian manifold in which the signature of the metric is (1, n−1) (equivalently, (n−1, 1); see Sign convention). Such metrics are called Lorentzian metrics. They are named after the Dutch physicist Hendrik Lorentz.

What is a Riemannian manifold in geometry?

In differential geometry, a Riemannian manifold or Riemannian space (M, g) is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p. A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n2 functions are smooth functions.

Can Riemannian metrics be less than smooth?

Riemannian metrics produced by methods of geometric analysis, in particular, can be less than smooth. See for instance (Gromov 1999) and (Shi and Tam 2002). Examples of Riemannian manifolds will be discussed below. A famous theorem of John Nash states that, given any smooth Riemannian manifold

How did Einstein use pseudo-Riemannian manifolds?

Albert Einstein used the theory of pseudo-Riemannian manifolds (a generalization of Riemannian manifolds) to develop his general theory of relativity. In particular, his equations for gravitation are constraints on the curvature of spacetime.

What is the tangent bundle of a smooth manifold?

Overview. The tangent bundle of a smooth manifold M assigns to each fixed point of M a vector space called the tangent space, and each tangent space can be equipped with an inner product. If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold,…