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3-sphere

The 3-sphere, denoted S^3, is a three-dimensional manifold defined as the set of points (w, x, y, z) in four-dimensional Euclidean space \mathbb{R}^4 satisfying w^2 + x^2 + y^2 + z^2 = r^2, where r > 0 is the radius (with the unit 3-sphere corresponding to r = 1).^[1] This hypersurface generalizes the ordinary 2-sphere S^2 (the surface of a ball in \mathbb{R}^3) to one higher dimension, serving as the boundary of the 4-dimensional ball in \mathbb{R}^4.^[2] Topologically, S^3 is compact, connected, and simply connected, with trivial fundamental group \pi_1(S^3) = 0, making it the unique simply connected 3-manifold up to homeomorphism by the Poincaré conjecture (now theorem).^[3] Its homology groups are H_0(S^3; \mathbb{Z}) = \mathbb{Z}, H_1(S^3; \mathbb{Z}) = 0, H_2(S^3; \mathbb{Z}) = 0, and H_3(S^3; \mathbb{Z}) = \mathbb{Z}, reflecting its sphere-like structure in higher dimensions.^[3] The 3-sphere admits various coordinate parameterizations, such as hyperspherical coordinates or the Hopf coordinates (\eta, \xi_1, \xi_2), which facilitate its study and reveal fibrations like the Hopf fibration mapping S^3 onto S^2 with S^1 fibers.^[1] A notable feature of S^3 is its identification with the group of unit quaternions, which endows it with a natural Lie group structure isomorphic to the special unitary group SU(2) of 2×2 complex unitary matrices with determinant 1.^[4] This isomorphism arises from representing unit quaternions q = a + bi + cj + dk (with a^2 + b^2 + c^2 + d^2 = 1) as matrices \begin{pmatrix} a + bi & c + di \\ -c + di & a - bi \end{pmatrix}, preserving the group operation of quaternion multiplication.^[4] Consequently, S^3 acts as a double cover of the rotation group SO(3) via the adjoint representation, where the kernel is \{1, -1\}, linking it to three-dimensional rotations and applications in physics, such as spin representations.^[4] Geometrically, the "surface area" (3-dimensional volume) of the unit 3-sphere is $2\pi^2, scaling as $2\pi^2 r^3 for general radius r, derived from integrating over 4-dimensional spherical coordinates or as the derivative of the enclosed 4-ball volume \frac{1}{2} \pi^2 r^4.^[5] These properties make S^3 central to differential geometry, topology, and quantum mechanics, where it models spaces like the configuration space of rigid body rotations.^[2]

Definition

Implicit equation

The 3-sphere, denoted S^3, is defined as the set of points (w, x, y, z) in \mathbb{R}^4 satisfying the equation

w^2 + x^2 + y^2 + z^2 = 1.

^[2] This represents the unit 3-sphere, consisting of all points at Euclidean distance 1 from the origin in four-dimensional space. More generally, the 3-sphere is a special case of the n-sphere S^n, defined as the set \{ \mathbf{x} \in \mathbb{R}^{n+1} : \| \mathbf{x} \| = 1 \}, where \| \cdot \| denotes the Euclidean norm.^[2] This embedding in \mathbb{R}^{n+1} generalizes the familiar 2-sphere S^2, which is the set of points (x, y, z) in \mathbb{R}^3 satisfying x^2 + y^2 + z^2 = 1, forming the surface of a ball in three dimensions.^[2] The equation for S^3 follows analogously by extending to one additional dimension, capturing the "surface" of the unit ball in \mathbb{R}^4. The hypersurface measure (analogous to surface area) of the unit 3-sphere is $2\pi^2.^[2] For a 3-sphere of radius r, this scales to $2\pi^2 r^3.^[2]

Manifold structure

The 3-sphere S^3 is a compact, connected, and orientable 3-dimensional manifold without boundary, serving as the standard model for simply connected closed 3-manifolds in differential topology.^[3] As an abstract smooth manifold, S^3 admits a smooth structure defined by an atlas of coordinate charts, which can be constructed using stereographic projections from points on the manifold to \mathbb{R}^3, ensuring compatibility of transition maps that are smooth diffeomorphisms.^[6] This atlas covers S^3 with two charts (or more for refinement), excluding antipodal points for each projection, and establishes S^3 as a smooth manifold diffeomorphic to the boundary of the 4-dimensional unit ball in \mathbb{R}^4.^[7] Equipped with the standard embedding in \mathbb{R}^4 defined by the equation w^2 + x^2 + y^2 + z^2 = 1, S^3 inherits a Riemannian metric as the restriction of the Euclidean metric on \mathbb{R}^4, given by

ds^2 = dw^2 + dx^2 + dy^2 + dz^2

on the tangent spaces to the hypersurface.^[8] This induced metric renders S^3 a Riemannian manifold of constant sectional curvature 1 (for the unit sphere), making it the unique simply connected space form in dimension 3 with positive constant curvature.^[8]^[9] Geodesics on S^3 are great circles, which are intersections with 2-planes through the origin in \mathbb{R}^4. The 3-dimensional volume (or content) of the unit 3-sphere is $2\pi^2, computed via integration over the hypersurface measure induced by the Riemannian metric.^[10] For a 3-sphere of radius r, this scales to $2\pi^2 r^3, reflecting its homogeneity and the uniformity of the metric.^[10]

Properties

Topological properties

The 3-sphere S^3 is simply connected, meaning its fundamental group \pi_1(S^3) is trivial.

\] This property follows from the general fact that the fundamental group of any $n$-sphere with $n \geq 2$ vanishes, as loops in $S^3$ can be contracted to a point without obstruction.\[

Consequently, S^3 has no non-trivial 1-dimensional holes, distinguishing it from spaces like the circle S^1, where \pi_1(S^1) \cong \mathbb{Z}. The higher homotopy groups of S^3 provide further insight into its topological structure. Specifically, \pi_k(S^3) is trivial for k < 3, \pi_3(S^3) \cong \mathbb{Z}, and for k > 3, the groups stabilize and match the stable homotopy groups of spheres, which are generally finite or include \mathbb{Z} summands in certain dimensions (e.g., \pi_4(S^3) \cong \mathbb{Z}/2\mathbb{Z}).

\] The [isomorphism](/page/Isomorphism) $\pi_3(S^3) \cong \mathbb{Z}$ arises from the [degree](/page/Degree) of maps from $S^3$ to itself, capturing the [winding number](/page/Winding_number) of such maps, while the trivial lower groups confirm the absence of lower-dimensional obstructions.\[

These homotopy groups underscore S^3's role as a basic building block in algebraic topology, with the Hopf fibration S^1 \to S^3 \to S^2 illustrating the non-triviality at dimension 3. In terms of homology, the singular homology groups of S^3 with integer coefficients are H_0(S^3) \cong \mathbb{Z}, H_3(S^3) \cong \mathbb{Z}, and H_k(S^3) = 0 for all other k \geq 1.

\] The $\mathbb{Z}$ in dimension 0 reflects the connectedness of $S^3$, while the generator in dimension 3 corresponds to the fundamental class, encoding the [orientability](/page/Orientability) and [compactness](/page/Compact_space) of the manifold.\[

By the Hurewicz theorem, since \pi_1(S^3) and \pi_2(S^3) are trivial, the first non-vanishing homology group aligns with \pi_3(S^3), linking homotopy and homology invariants. The Poincaré conjecture, resolved by Grigori Perelman's proof in 2002–2003 using Ricci flow with surgery, asserts that every simply connected, closed 3-manifold is homeomorphic to S^3, making S^3 the unique such manifold up to homeomorphism.

\] This result, building on Richard Hamilton's program, implies that $S^3$ cannot be deformed topologically into any other simply connected 3-manifold without singularities.\[

As a simply connected space, the universal cover of S^3 is S^3 itself, with the identity map serving as the covering projection.$$] This self-covering property reinforces S^3's minimal topological complexity among 3-manifolds.

Geometric properties

The unit 3-sphere S^3, embedded in \mathbb{R}^4 with the induced round metric, possesses constant sectional curvature equal to 1.^[2] Its geodesics are great circles, defined as the intersections of S^3 with 2-dimensional linear subspaces through the origin in \mathbb{R}^4. These great circles, being closed curves of minimal length connecting any two points, have a total circumference of $2\pi.^[2]^[11] The geodesic distance d(p, q) between two points p, q \in S^3 on the unit 3-sphere is the length of the great circle arc connecting them, given by d(p, q) = \arccos(p \cdot q) or equivalently d(p, q) = 2 \arcsin(\|p - q\| / 2). This metric arises from the Riemannian structure and ensures that the space is complete and simply connected, supporting unique minimizing geodesics between points. The maximum such distance, or diameter of S^3, is \pi, achieved between antipodal points.^[2] The full group of orientation-preserving isometries of the unit 3-sphere is SO(4), which acts transitively on S^3, reflecting its high degree of symmetry as a homogeneous space. Including reflections, the complete isometry group is O(4).^[12] Viewing S^3 as a hypersurface in \mathbb{R}^4, it has three equal principal curvatures of 1. The unit 3-sphere has 3-dimensional volume $2\pi^2. For context, the 2-sphere S^2 (unit radius) has surface area $4\pi with constant Gaussian curvature 1, whereas the 3-sphere has 3-dimensional volume $2\pi^2, highlighting the progression in hyperspherical measures across dimensions.^[2]

Constructions

Gluing construction

The 3-sphere S^3 can be constructed topologically as the union of two solid tori, each homeomorphic to D^2 \times S^1, glued along their common boundary torus T^2 = S^1 \times S^1.^[3] A solid torus is formed by taking the product of a 2-dimensional disk D^2 and a circle S^1, where the boundary consists of the product of the boundary circle \partial D^2 = S^1 (the meridian) and S^1 (the longitude).^[13] This decomposition arises naturally from viewing S^3 as the boundary of the 4-dimensional ball D^4, which is equivalently \partial(D^2 \times D^2) = (\partial D^2 \times D^2) \cup (D^2 \times \partial D^2), yielding the two solid tori via the identity identification on the boundary.^[3] The gluing process proceeds by attaching the boundaries of the two solid tori via an orientation-reversing homeomorphism that swaps the roles of the meridian and longitude: specifically, the meridional circle of the first solid torus is identified with the longitudinal circle of the second, and vice versa.^[13] This identification ensures that the boundary tori are glued pointwise without twisting, preserving the overall topology. To visualize this, consider the core circles of the two solid tori—the S^1 factors at the centers of the D^2s—which become linked in the resulting space, forming the Hopf link, the simplest non-trivial link in S^3.^[13] This construction corresponds to the genus 1 Heegaard splitting of S^3, where the splitting surface is the embedded torus separating the two solid tori.^[3] Among closed orientable 3-manifolds, S^3 is distinguished by having a unique genus 1 Heegaard splitting up to isotopy, as established by Waldhausen's theorem on the uniqueness of Heegaard splittings for the 3-sphere.^[14] Higher-genus splittings of S^3 arise via stabilization of this genus 1 surface, but the minimal non-trivial toroidal decomposition is this genus 1 case.^[15] To verify that this gluing yields S^3, consider a proof sketch using the Seifert–van Kampen theorem on the fundamental group. Each solid torus has fundamental group \mathbb{Z} generated by its longitude, while the boundary torus has \pi_1(T^2) = \mathbb{Z} \times \mathbb{Z} generated by the meridian \mu and longitude \lambda. Let \mu_1, \lambda_1 be the meridian and longitude of the first solid torus, and \mu_2, \lambda_2 for the second. The inclusion of the boundary into each solid torus sends the meridian to the trivial element (as it bounds a disk) but the longitude to the generator. The gluing map identifies \mu_1 with \lambda_2 and \lambda_1 with \mu_2, so the amalgamated product imposes relations that set \lambda_2 = 1 (from \mu_1 = 1) and \lambda_1 = 1 (from \mu_2 = 1), making the fundamental group trivial and confirming \pi_1(S^3) = 0.^[13] This simply connectedness aligns with the known topological properties of S^3.^[3]

Compactification methods

The 3-sphere arises as the one-point compactification of Euclidean 3-space \mathbb{R}^3, obtained by adjoining a single point \infty to \mathbb{R}^3 and equipping the resulting space \mathbb{R}^3 \cup \{\infty\} with the topology where the open sets are the open subsets of \mathbb{R}^3 together with sets of the form \mathbb{R}^3 \setminus K \cup \{\infty\} for compact K \subset \mathbb{R}^3. This construction yields a compact Hausdorff space homeomorphic to S^3, as \mathbb{R}^3 is locally compact and non-compact.^[16] The neighborhoods of the point at infinity thus consist of complements of compact sets in \mathbb{R}^3, capturing the behavior at spatial infinity.^[16] This one-point compactification coincides with the Alexandrov compactification for \mathbb{R}^3, as the latter is defined identically for locally compact Hausdorff spaces by adding a point whose neighborhoods are complements of compact subsets. A concrete realization of this compactification uses stereographic projection, which establishes a homeomorphism between \mathbb{R}^3 and S^3 minus one point (typically the north pole (1,0,0,0)). Under this map, points x = (x_1, x_2, x_3, x_4) \in S^3 \setminus \{(1,0,0,0)\} project to \left( \frac{x_2}{1 - x_1}, \frac{x_3}{1 - x_1}, \frac{x_4}{1 - x_1} \right) \in \mathbb{R}^3, with the excluded point corresponding to infinity. This projection inverts distances near infinity, mapping large Euclidean distances to small distances on the sphere near the pole, thereby compactifying \mathbb{R}^3 conformally. Consequently, \mathbb{R}^3 is homeomorphic to S^3 minus a point via this inversion. In relativity, the 3-sphere features in the conformal compactification of Minkowski spacetime, where the compactified structure is conformally equivalent to an open dense subset of the Einstein static universe S^1 \times S^3, aiding analysis of null infinity and asymptotic flatness.^[17]

Coordinate Systems

Hyperspherical coordinates

The hyperspherical coordinates offer a natural parameterization of the 3-sphere S^3 as a subset of \mathbb{R}^4, extending the spherical coordinates used for lower-dimensional spheres. These coordinates use three angular variables to describe points on the unit 3-sphere, satisfying the equation w^2 + x^2 + y^2 + z^2 = 1. Specifically, a point (w, x, y, z) \in S^3 is given by [ \begin{align*} w &= \cos \chi, \ x &= \sin \chi \cos \theta, \ y &= \sin \chi \sin \theta \cos \phi, \ z &= \sin \chi \sin \theta \sin \phi, \end{align*}

where the ranges are $\chi \in [0, \pi]$, $\theta \in [0, \pi]$, and $\phi \in [0, 2\pi)$.[](https://mathworld.wolfram.com/Hypersphere.html)[](https://arxiv.org/pdf/1902.00285) The induced Riemannian metric on $S^3$ in these coordinates takes the form

ds^2 = d\chi^2 + \sin^2 \chi \left( d\theta^2 + \sin^2 \theta , d\phi^2 \right).

This metric reflects the geometry of nested spheres: the $\chi$-direction corresponds to the "radial" angle in the embedding space, while the terms in parentheses describe the standard metric on the 2-sphere of radius $\sin \chi$. The corresponding [volume element](/page/Volume_element), or Jacobian for integration, is $dV = \sin^2 \chi \sin \theta \, d\chi \, d\theta \, d\phi$.[](https://mathworld.wolfram.com/Hypersphere.html) The coordinate ranges introduce singularities at specific values, analogous to poles and meridians on a 2-sphere. At $\chi = 0$ and $\chi = \pi$, the hypersurface collapses to single points (the "poles"), where $\sin \chi = 0$ and the $\theta$-$\phi$ spheres degenerate. Similarly, at $\theta = 0$ and $\theta = \pi$, the $\phi$-circles reduce to lines, causing coordinate degeneracies. These points require careful handling in computations, often by excluding them or using alternative charts.[](https://arxiv.org/pdf/1902.00285) As an example of [integration](/page/Integration) in these coordinates, the 3-dimensional [volume](/page/Volume) (or "surface area" in [4D](/page/4D)) of the unit 3-sphere is obtained by evaluating

\int_0^\pi \int_0^\pi \int_0^{2\pi} \sin^2 \chi \sin \theta , d\phi , d\theta , d\chi = 2\pi^2.

This result follows from separating the integrals: $\int_0^{2\pi} d\phi = 2\pi$, $\int_0^\pi \sin \theta \, d\theta = 2$, and $\int_0^\pi \sin^2 \chi \, d\chi = \pi/2$.[](https://mathworld.wolfram.com/Hypersphere.html) ### Hopf coordinates Hopf coordinates parametrize the 3-sphere $S^3$ in a manner that explicitly reveals its structure as a circle bundle over the 2-sphere $S^2$ through the [Hopf fibration](/page/Hopf_fibration) $S^1 \to S^3 \to S^2$. These coordinates consist of $\eta \in [0, \pi/2]$, $\psi \in [0, 2\pi)$, $\phi \in [0, 2\pi)$, where the embedding is given in complex coordinates as $z_1 = \cos \eta \, e^{i \psi}$, $z_2 = \sin \eta \, e^{i \phi}$ (with $w + i x = z_1$, $y + i z = z_2$), corresponding to the unit quaternion representation.[](https://arxiv.org/pdf/1505.03426) The induced metric on the unit $S^3$ in these coordinates is

ds^2 = d\eta^2 + \cos^2 \eta , d\psi^2 + \sin^2 \eta , d\phi^2.

[](https://arxiv.org/pdf/1505.03426) The coordinates are orthogonal, as the [metric tensor](/page/Metric_tensor) is diagonal in this basis, facilitating computations of geodesics and volumes. The [Hopf fibration](/page/Hopf_fibration) projection maps points on $S^3$ to the base $S^2$ via the coordinates $(\eta, \psi - \phi)$, where each fiber over a base point is an $S^1$ [circle](/page/Circle) corresponding to the common phase shift.[](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/hopf.pdf) The volume element on $S^3$ is $\sin \eta \cos \eta \, d\eta \, d\phi \, d\psi$, yielding the total volume $\int_0^{\pi/2} \sin \eta \cos \eta \, d\eta \int_0^{2\pi} d\phi \int_0^{2\pi} d\psi = 2\pi^2$.[](https://arxiv.org/pdf/1505.03426) These coordinates relate to the discovery of the [Hopf fibration](/page/Hopf_fibration) by [Heinz Hopf](/page/Heinz_Hopf) in his 1931 paper, providing the foundational framework for understanding the nontrivial [topology](/page/Topology) of the bundle.[](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/hopf.pdf) ### Stereographic coordinates The [stereographic projection](/page/Stereographic_projection) provides a [diffeomorphism](/page/Diffeomorphism) from the 3-sphere minus the [north pole](/page/North_Pole) to [Euclidean](/page/Euclidean) 3-space $\mathbb{R}^3$, serving as a local coordinate chart for the manifold.[](https://dspace.mit.edu/bitstream/handle/1721.1/45589/18-701Fall2003/NR/rdonlyres/Mathematics/18-701Fall2003/9C8896A9-B6C9-49C1-BF54-FC52C5DC0121/0/herm_geom03.pdf) Considering points on the unit 3-sphere $S^3 = \{(w, x, y, z) \in \mathbb{R}^4 \mid w^2 + x^2 + y^2 + z^2 = 1\}$, the projection from the [north pole](/page/North_Pole) $(1, 0, 0, 0)$ maps a point $(w, x, y, z)$ to coordinates $(u, v, t) \in \mathbb{R}^3$ via

u = \frac{x}{1 - w}, \quad v = \frac{y}{1 - w}, \quad t = \frac{z}{1 - w}.

[](https://arxiv.org/pdf/0908.2114) This mapping is undefined at the [north pole](/page/North_Pole), where $w = 1$ would make the denominator zero, corresponding to the point at [infinity](/page/Infinity) in $\mathbb{R}^3$.[](https://dspace.mit.edu/bitstream/handle/1721.1/45589/18-701Fall2003/NR/rdonlyres/Mathematics/18-701Fall2003/9C8896A9-B6C9-49C1-BF54-FC52C5DC0121/0/herm_geom03.pdf) The inverse mapping recovers points on $S^3$ from $(u, v, t) \in \mathbb{R}^3$, where $r^2 = u^2 + v^2 + t^2$. It is given by

w = \frac{1 - r^2}{1 + r^2}, \quad x = \frac{2u}{1 + r^2}, \quad y = \frac{2v}{1 + r^2}, \quad z = \frac{2t}{1 + r^2}.

[](https://dspace.mit.edu/bitstream/handle/1721.1/45589/18-701Fall2003/NR/rdonlyres/Mathematics/18-701Fall2003/9C8896A9-B6C9-49C1-BF54-FC52C5DC0121/0/herm_geom03.pdf) This formula ensures that the image lies on $S^3$, as substituting yields $w^2 + x^2 + y^2 + z^2 = 1$, and it covers all points except the [north pole](/page/North_Pole).[](https://arxiv.org/pdf/0908.2114) The pullback of the round [metric](/page/Metric) on $S^3$ under this [projection](/page/Projection) yields a conformal [metric](/page/Metric) on $\mathbb{R}^3$:

ds^2 = \frac{4 (du^2 + dv^2 + dt^2)}{(1 + r^2)^2}.

[](https://brianwebermathematics.com/wp-content/uploads/2018/11/spheresprimer4.pdf) This [metric](/page/Metric) is conformal to the [Euclidean](/page/Euclidean) [metric](/page/Metric) with conformal factor $4 / (1 + r^2)^2$, preserving angles and making $\mathbb{R}^3$ a model for [local](/page/.local) [geometry](/page/Geometry) near any point except the projection [pole](/page/Pole).[](https://arxiv.org/pdf/0908.2114) The conformal [property](/page/Property) facilitates computations in [differential geometry](/page/Differential_geometry) and analysis on $S^3$.[](https://brianwebermathematics.com/wp-content/uploads/2018/11/spheresprimer4.pdf) Viewing $S^3$ as the unit sphere in $\mathbb{C}^2$, the stereographic coordinates align with complex structures, enabling applications in [complex analysis](/page/Complex_analysis) by mapping to $\mathbb{C}^2$ minus a point.[](https://dspace.mit.edu/bitstream/handle/1721.1/45589/18-701Fall2003/NR/rdonlyres/Mathematics/18-701Fall2003/9C8896A9-B6C9-49C1-BF54-FC52C5DC0121/0/herm_geom03.pdf) This perspective is useful for studying holomorphic functions and extensions across the compactification.[](https://dspace.mit.edu/bitstream/handle/1721.1/45589/18-701Fall2003/NR/rdonlyres/Mathematics/18-701Fall2003/9C8896A9-B6C9-49C1-BF54-FC52C5DC0121/0/herm_geom03.pdf) The projection has a singularity at the north pole, so a single chart does not cover all of $S^3$; instead, an atlas with at least two charts (e.g., from opposite poles) is required to provide global coordinates.[](https://arxiv.org/pdf/0908.2114) This atlas ensures $S^3$ is a [smooth](/page/Smooth) manifold, with transition maps being diffeomorphisms between overlapping regions in $\mathbb{R}^3$.[](https://dspace.mit.edu/bitstream/handle/1721.1/45589/18-701Fall2003/NR/rdonlyres/Mathematics/18-701Fall2003/9C8896A9-B6C9-49C1-BF54-FC52C5DC0121/0/herm_geom03.pdf) ## Algebraic Aspects ### Lie group structure The 3-sphere $ S^3 $, defined as the set of points $ (w,x,y,z) \in \mathbb{R}^4 $ with $ w^2 + x^2 + y^2 + z^2 = 1 $, admits a natural [Lie group](/page/Lie_group) structure by identifying it with the [multiplicative group](/page/Multiplicative_group) of unit quaternions, which is diffeomorphic to the [special unitary group](/page/Special_unitary_group) $ \mathrm{SU}(2) $. This identification equips $ S^3 $ with a smooth manifold structure compatible with the group operation of quaternion multiplication, making it a compact, connected, and simply connected [Lie group](/page/Lie_group) of dimension 3.[](https://www.math.stonybrook.edu/~sunscorch/teaching/Lie_Theory.pdf) The group law on unit quaternions $ q = w + x\mathbf{i} + y\mathbf{j} + z\mathbf{k} $ with $ |q| = 1 $ is given by the standard quaternion multiplication, preserving the unit norm and ensuring the operation is associative and invertible.[](https://www.cis.upenn.edu/~cis5150/gma-v2-chap9.pdf) The [Lie algebra](/page/Lie_algebra) of $ S^3 \cong \mathrm{SU}(2) $ is $ \mathfrak{su}(2) $, the real [vector space](/page/Vector_space) of $ 2 \times 2 $ skew-Hermitian traceless [complex](/page/Complex) matrices, which is isomorphic to $ \mathfrak{so}(3) $ as [Lie algebras](/page/Lie_algebra). A [standard basis](/page/Standard_basis) for $ \mathfrak{su}(2) $ consists of $ i $ times the [Pauli matrices](/page/Pauli_matrices):

\sigma_1 = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}, \quad \sigma_3 = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix},

so the basis elements are $ i\sigma_1, i\sigma_2, i\sigma_3 $, with the [Lie bracket](/page/Lie_algebra) corresponding to the [cross product](/page/Cross_product) in $ \mathbb{R}^3 $.[](https://www.cis.upenn.edu/~cis5150/gma-v2-chap9.pdf) This [isomorphism](/page/Isomorphism) $ \mathfrak{su}(2) \cong \mathfrak{so}(3) $ reflects the shared structure of infinitesimal rotations in three dimensions. The [adjoint representation](/page/Adjoint_representation) of $ \mathrm{SU}(2) $ on its [Lie algebra](/page/Lie_algebra) induces a surjective [Lie group](/page/Lie_group) homomorphism $ \mathrm{SU}(2) \to \mathrm{SO}(3) $ with kernel $ \{\pm I\} $, establishing $ S^3 $ as a double cover of the rotation group $ \mathrm{SO}(3) $.[](https://www.math.stonybrook.edu/~sunscorch/teaching/Lie_Theory.pdf) The [exponential map](/page/Exponential_map) $ \exp: \mathfrak{su}(2) \to \mathrm{SU}(2) $ is a surjective [diffeomorphism](/page/Diffeomorphism) for this compact connected [Lie group](/page/Lie_group), explicitly given for a pure [imaginary unit](/page/Imaginary_unit) [quaternion](/page/Quaternion) $ v $ (with $ |v| = 1 $) by $ \exp(\theta v) = \cos(\theta/2) + \sin(\theta/2) v $, which traces geodesics on $ S^3 $ and parameterizes the group elements efficiently.[](https://www.cis.upenn.edu/~cis5150/gma-v2-chap9.pdf) As a compact [Lie group](/page/Lie_group), $ S^3 $ possesses a unique (up to positive scalar multiple) bi-invariant [Haar measure](/page/Haar_measure), which can be normalized such that the total volume is $ 2\pi^2 $; in hyperspherical coordinates, this measure takes the form proportional to $ \sin^2(\theta) \sin(\psi) \, d\theta \, d\psi \, d\phi $.[](https://web.math.ku.dk/~jakobsen/replie/compact-work.pdf) The full isometry group of $ S^3 $ embedded in $ \mathbb{R}^4 $ with the round metric is $ \mathrm{SO}(4) $, acting transitively on the sphere. This group decomposes as $ \mathrm{SO}(4) \cong (\mathrm{SU}(2) \times \mathrm{SU}(2))/\mathbb{Z}_2 $, where the quotient identifies $ (g, h) $ with $ (-g, -h) $ to account for the kernel of the double cover map from $ \mathrm{SU}(2) \times \mathrm{SU}(2) $ to $ \mathrm{SO}(4) $, reflecting the left and right multiplications by unit quaternions.[](https://www.physik.uni-wuerzburg.de/fileadmin/11030200/Personen_Ohl/Lehre/TPP/2015/tpp-07.pdf) ### Quaternion representation The 3-sphere $ S^3 $ is algebraically realized as the set of unit quaternions, which are elements $ q = w + x i + y j + z k $ in the quaternion algebra $ \mathbb{H} $ over $ \mathbb{R} $ satisfying the unit norm condition $ \|q\|^2 = w^2 + x^2 + y^2 + z^2 = 1 $.[](https://www.cis.upenn.edu/~cis6100/geombchap8.pdf)[](https://math.umd.edu/~wmg/quaternions.pdf) This identification embeds $ S^3 $ as a hypersurface in the 4-dimensional real vector space underlying $ \mathbb{H} $, where the basis elements satisfy $ i^2 = j^2 = k^2 = -1 $ and $ ij = k $, $ jk = i $, $ ki = j $.[](https://math.umd.edu/~wmg/quaternions.pdf) Quaternion multiplication is bilinear and non-commutative, with the product of two [quaternions](/page/Quaternion) $ q_1 = a_1 + \mathbf{u}_1 $ and $ q_2 = a_2 + \mathbf{u}_2 $ (where $ a $ is the real part and $ \mathbf{u} $ the vector part) given by $ q_1 q_2 = (a_1 a_2 - \mathbf{u}_1 \cdot \mathbf{u}_2) + (a_1 \mathbf{u}_2 + a_2 \mathbf{u}_1 + \mathbf{u}_1 \times \mathbf{u}_2) $.[](https://math.umd.edu/~wmg/quaternions.pdf)[](https://www.cis.upenn.edu/~cis6100/geombchap8.pdf) The conjugation operation $ \overline{q_1 q_2} = \overline{q_2} \overline{q_1} $ is anti-multiplicative, and the norm is multiplicative via $ \|q_1 q_2\| = \|q_1\| \|q_2\| $, ensuring that the product of unit quaternions remains a unit quaternion and thus preserves the geometry of $ S^3 $.[](https://www.cis.upenn.edu/~cis6100/geombchap8.pdf)[](https://math.umd.edu/~wmg/quaternions.pdf) This multiplicative structure endows $ S^3 $ with the topology of a [Lie group](/page/Lie_group), isomorphic to the [special unitary group](/page/Special_unitary_group) $ SU(2) $.[](https://www.cis.upenn.edu/~cis6100/geombchap8.pdf) Unit quaternions provide a double cover of the [rotation](/page/Rotation) group $ SO(3) $, where each [rotation](/page/Rotation) corresponds to a pair $ \{q, -q\} $, acting on pure imaginary quaternions ([identified](/page/Identified) with $ \mathbb{R}^3 $) via the conjugation map $ v \mapsto q v \overline{q} $ for a pure quaternion $ v $.[](https://www.cs.unc.edu/~dm/UNC/COMP236/papers/quat.pdf)[](https://www.cis.upenn.edu/~cis6100/geombchap8.pdf) This representation is efficient for composing [rotations](/page/Rotation), as the group operation on quaternions corresponds to [matrix multiplication](/page/Matrix_multiplication) in $ SO(3) $. Any unit quaternion admits an exponential form $ q = \cos(\theta/2) + \sin(\theta/2) \, u $, where $ u $ is a unit pure imaginary quaternion representing the [rotation](/page/Rotation) [axis](/page/Axis) and $ \theta $ the angle, facilitating the parametrization of [rotations](/page/Rotation) around arbitrary [axes](/page/Axis).[](https://math.umd.edu/~wmg/quaternions.pdf)[](https://www.cs.unc.edu/~dm/UNC/COMP236/papers/quat.pdf) In computational applications, particularly 3D graphics and [animation](/page/Animation), unit quaternions enable smooth [interpolation](/page/Interpolation) of rotations via spherical linear [interpolation](/page/Interpolation) (SLERP), which traces the shortest [geodesic](/page/Geodesic) path on $ S^3 $ between two quaternions $ q_0 $ and $ q_1 $:

\text{SLERP}(q_0, q_1, t) = \frac{\sin((1-t)\theta)}{\sin \theta} q_0 + \frac{\sin(t \theta)}{\sin \theta} q_1,

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