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The above -sphere exists in (+) -dimensional Euclidean space and is an example of an -manifold. The volume form ω {\displaystyle \omega } of an n {\displaystyle n} -sphere of radius r {\displaystyle r} is given by
The circle can be represented by a graph in the neighborhood of every point because the left hand side of its defining equation + = has nonzero gradient at every point of the circle. By the implicit function theorem , every submanifold of Euclidean space is locally the graph of a function.
More generally, stereographic projection may be applied to the unit n-sphere S n in (n + 1)-dimensional Euclidean space E n+1. If Q is a point of S n and E a hyperplane in E n+1, then the stereographic projection of a point P ∈ S n − {Q} is the point P ′ of intersection of the line QP with E.
The n dimensional real projective space is the quotient of the n sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere – either 0 or 1. The n dimensional torus is the product space of n circles. Its Euler characteristic is 0, by the product property.
In an isotropic chart (on a static spherically symmetric spacetime), the metric (aka line element) takes the form = + (+ (+ ())), < <, < <, < <, < < Depending on context, it may be appropriate to regard , as undetermined functions of the radial coordinate (for example, in deriving an exact static spherically symmetric solution of the Einstein field equation).
By selecting this open set to be contained in a coordinate chart, one can reduce the claim to the well-known fact that, in Euclidean geometry, the shortest curve between two points is a line. In particular, as seen by the Euclidean geometry of a coordinate chart around p , any curve from p to q must first pass though a certain "inner radius."
A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. [2]
When at most three regions meet at a point, the result is a planar graph, but when four or more regions meet at a point, the result can be nonplanar (for example, if one thinks of a circle divided into sectors, with the sectors being the regions, then the corresponding map graph is the complete graph as all the sectors have a common boundary ...