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This mapping is known as a Riemann mapping. [1] Intuitively, the condition that be simply connected means that does not contain any “holes”. The fact that is biholomorphic implies that it is a conformal map and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly ...
exponential map (Riemannian geometry) for a manifold with a Riemannian metric, exponential map (Lie theory) from a Lie algebra to a Lie group, More generally, in a manifold with an affine connection, (), where is a geodesic with initial velocity X, is sometimes also called the exponential map. The above two are special cases of this with ...
The exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography. In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical ...
In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let M be a Riemannian manifold, equipped with its Levi-Civita connection, and p a point of M. The exponential map is a mapping from the tangent space ...
A 1-parameter group of conformal transformations gives rise to a vector field X with the property that the Lie derivative of g along X is proportional to g. Symbolically, L X g = λg for some λ. In particular, using the above description of the Lie algebra cso(1, 1), this implies that L X dx = a(x) dx; L X dy = b(y) dy
A harmonic map heat flow on an interval (a, b) assigns to each t in (a, b) a twice-differentiable map f t : M → N in such a way that, for each p in M, the map (a, b) → N given by t ↦ f t (p) is differentiable, and its derivative at a given value of t is, as a vector in T f t (p) N, equal to (∆ f t ) p. This is usually abbreviated as:
The mapping class group of is the coset group () of the diffeomorphism group of by the normal subgroup of those that are isotopic to the identity (the same definition can be made with homeomorphisms instead of diffeomorphisms and, for surfaces, this does not change the resulting group). The group of diffeomorphisms acts naturally on ...
This demonstrates that a Riemannian metric and an orientation on a two-dimensional manifold combine to induce the structure of a Riemann surface (i.e. a one-dimensional complex manifold). Furthermore, given an oriented surface, two Riemannian metrics induce the same holomorphic atlas if and only if they are conformal to one another.