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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
For example, the class of two-dimensional Euclidean space forms includes Riemannian metrics on the Klein bottle, the Möbius strip, the torus, the cylinder S 1 × ℝ, along with the Euclidean plane. Unlike the case of two-dimensional spherical space forms, in some cases two space form structures on the same manifold are not homothetic.
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).
A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation + + + = defines an algebraic hypersurface of dimension n − 1 in the Euclidean space of dimension n.
An embedding of the Euclidean sphere into N +, as in the previous section, determines a conformal scale on S. Conversely, any conformal scale on S is given by such an embedding. Thus the line bundle N + → S is identified with the bundle of conformal scales on S : to give a section of this bundle is tantamount to specifying a metric in the ...
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.The sectional curvature K(σ p) depends on a two-dimensional linear subspace σ p of the tangent space at a point p of the manifold.
In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space. More generally, the unit n {\displaystyle n} -sphere is an n {\displaystyle n} -sphere of unit radius in ( n + 1 ) {\displaystyle (n+1)} - dimensional Euclidean space ; the unit circle is a ...
A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a ...