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A Lie group is a group which is a finite-dimensional smooth Hausdorff second-countable manifold, with smooth multiplication and inversion. For the special unitary group, consider the real vector spaceThe diagonal entries of a skew-Hermitian matrix supply $n$ real parameters, and the entries above the diagonal supply $2\left(\genfrac{}{}{0ex}{}{n}{2}\right)$; the trace-zero condition removes one real parameter. Thus ${\dim }_{\mathbb{R}}\mathfrak{su}(n)=n^2-1$.

The matrix facts used to construct charts are these: the matrix exponential is analytic, its derivative at zero is the identity, and it has an analytic inverse matrix logarithm on a sufficiently small neighborhood of the identity. On those neighborhoods $\log (e^X)=X$, $e^{\log U}=U$, and $\det (e^X)=e^{\mathrm{tr}X}$. For a unitary matrix in that neighborhood, the spectral theorem writes $U=W\mathrm{diag}(e^{i{\theta }_j})W^{\ast }$ with principal angles close to zero, and $\log U=W\mathrm{diag}(i{\theta }_j)W^{\ast }$ is skew-Hermitian.

Choose a norm ball of radius $\epsilon <\pi /n$ small enough to lie in the exponential's inverse domain. If $U$ is unitary, $\det U=1$ and $\Vert \log U\Vert <\epsilon$ in operator norm, thenso its trace is zero. Conversely $X\in \mathfrak{su}(n)$ gives $(e^X{)}^{\ast }{e}^X=I$ and $\det (e^X)=1$. Thus the logarithm identifies an open neighborhood $V$ of $I$ in $SU(n)$, with its matrix subspace topology, with the open ball $B_{\epsilon }\subset \mathfrak{su}(n)$. Choosing a real linear identification $\mathfrak{su}(n)\cong {\mathbb{R}}^{n^2-1}$ turns this into a chart.

For every $A\in SU(n)$ take the translated chartTheir overlap maps are $X\mapsto \log (B^{-1}A{e}^X)$ wherever both charts are defined, so they are real analytic. They cover the group; the matrix topology is Hausdorff and second-countable. This constructs the real manifold directly, rather than merely counting equations. Matrix multiplication is polynomial in real/imaginary entries and inversion here is $U\mapsto U^{\ast }$; in the logarithm charts both are smooth. The logarithm charts for the special unitary group therefore proveFor $n=1$ it is the one-point group of dimension zero.

For $n=2$, let the first column be $(a,b)$, of unit norm. Every unit vector orthogonal to it is a scalar of modulus one times $(-\bar{b},\bar{a})$. The determinant of the resulting matrix is that scalar, so determinant one fixes it to $1$. Henceis bijective, with inverse given by taking the first column. The forward map is real polynomial in the four coordinates; composing with the local logarithm charts verifies smoothness into the constructed manifold. The inverse is a smooth matrix-coordinate projection and, in the usual embedded sphere charts, is smooth into $S^3$. Thus $\boxed{SU(2)\cong S^3\text{ by a diffeomorphism}}$, the explicit realization of SU(2) as the three-sphere.

The topological degree assigns an integer to a continuous map $f:M\to N$ between closed, connected, oriented manifolds of the same dimension $d$. Their top homology groups are infinite cyclic, generated by their fundamental classes. The induced map definesFor a smooth map and a regular value $y$, its equivalent local description iswhere the sign records preservation or reversal of orientation. The preimage is a finite set because the manifolds are compact and $y$ is regular. This is a signed count; opposite-orientation preimages cancel. If $\omega$ is a normalized target volume form with ${\int }_N\omega =1$, the pullback of a differential form gives the equivalent integral $\mathrm{deg}f={\int }_M{f}^{\ast }\omega$.

The homotopy invariance of mapping degree follows because homotopic maps induce the same homology map. In the smooth description, apply Stokes theorem to a homotopy $F:M\times [0,1]\to N$:since a top-dimensional form on $N$ is closed. Thus degree cannot change during a smooth deformation preserving the relevant boundary conditions. Degrees multiply under composition, and a nonzero-degree map is surjective: a missed target point would give a regular value with an empty preimage and hence degree zero.

For maps $S^d\to S^d$, the homotopy group ${\pi }_d(S^d)=\mathbb{Z}$ is completely detected by degree. This supplies integer topological sectors for many classical field-theory solitons. It is important to distinguish two ways spheres arise. In a nonlinear sigma model with the field approaching one fixed target value at infinity, compactify all of ${\mathbb{R}}^d$ to $S^d$ and classify the resulting map into the target. For a defect whose vacuum field varies with direction, retain the sphere $S_{\infty }^{d-1}$ surrounding the core and map it into the vacuum manifold. This is the vacuum-boundary degree as a defect charge. An asymptotic limit is a boundary hypothesis; finite energy alone in every possible model need not imply one.

For an Abelian Higgs vortex, the phase at infinity defines $S^1\to U(1)\cong S^1$. Its degree is the winding numberNonzero $n$ forces a zero of the Higgs field inside: a nowhere-vanishing field would extend its normalized phase through the disk and make the boundary loop contractible. This is also the flux classification of the Bogomolny vortex equations.

For a 't Hooft-Polyakov monopole, the normalized adjoint Higgs field $\hat{\Phi }$ gives $S_{\infty }^2\to SU(2)/U(1)\cong S^2$. With outward orientation its degree isThe integrand is the normalized target area form, so the hedgehog ansatz for a monopole has degree one. Nonzero degree obstructs extending the normalized Higgs direction through the ball; the full Higgs field must leave the vacuum manifold, normally vanishing in its core. The asymptotic unbroken Abelian magnetic flux measures the same integer with a convention-dependent sign.

For the Skyrme model, a field $U:{\mathbb{R}}^3\to SU(2)$ with $U\to 1$ at infinity gives $S^3\to S^3$, because SU(2) as the three-sphere identifies the target. A choice of orientations gives its topological baryon number in the Skyrme model as the winding integral $\pm (24{\pi }^2{)}^{-1}\int \mathrm{tr}(U^{-1}dU{)}^3$; specifying the orientation fixes the sign. Similarly, the transition map at infinity of a four-dimensional Yang-Mills instanton has degree in ${\pi }_3(SU(2))$, measured by its Chern charge.

A scalar-field kink illustrates the discrete-vacuum case: its endpoint data lie in ${\pi }_0(\{\pm 1\})$, rather than being a map between connected positive-dimensional spheres. For the double-well field its topological charge is $Q=[\varphi (+\infty )-\varphi (-\infty )]/2$. A kink has $Q=1$, an antikink $Q=-1$; they cannot individually deform to vacuum while their endpoints are fixed.

Degree classifies topology, rather than every solution. Smooth motion or local perturbations with fixed boundary data preserve it, and a nonzero degree obstructs decay to the vacuum within that class. It does not guarantee existence of a smooth energy minimizer, a fixed size, uniqueness, or dynamical stability. The degree and energetic stability of a field configuration are distinct: a field can shrink towards a singular limit at fixed degree, as Derrick's theorem diagnoses in many two-derivative theories. Moreover degree need not classify all homotopy classes for an arbitrary target; other homotopy groups or invariants may be needed. Within one degree there can be continuous moduli spaces and several stationary solutions.

A Derrick scaling variation changes a configuration's size without changing its topological sector. Use $U_L(\mathbf{x})=U(\mathbf{x}/L)$, so $L>1$ expands the configuration. An integrated nonnegative term homogeneous in $s$ spatial derivatives scales as $L^{d-s}{E}_s$. For a scalar gradient-plus-potential energy, for example,A static solution is a stationary point of the full energy functional, so it must in particular satisfy $E^{\prime }(1)=0$. In this example the Derrick virial identity is $(d-2)E_2+d{E}_0=0$. If both energies are nonnegative and the coefficients cannot balance, a nontrivial localized static solution is impossible. If balance is possible, it fixes relations among energy contributions and may select a characteristic size. A negative $E^{\prime \prime }(1)$ proves an instability under size changes; a positive second derivative gives stability only against that one deformation. Passing the scale test is a necessary condition, not an existence theorem or a proof of full stability. It does not directly rule out time-dependent or charge-constrained solitons outside the static variational problem.

For the specified model, each spatial current is anti-Hermitian: differentiating $U{U}^{†}=1$ gives $R_i^{†}=-R_i$. Thus $-\mathrm{Tr}(R_i^2)$ is nonnegative, and the static energy of the nonlinear sigma model isThe boundary condition $U\to 1_2$ identifies spatial infinity to one point, so one-point compactification turns the field into a based map $S^d\to SU(2)$. By SU(2) as the three-sphere, its sectors are classified by the homotopy group ${\pi }_d(S^3)$. In particular,There is no corresponding topological protection in one or two spatial dimensions. In three dimensions there is an integer degree of a map between oriented manifolds, the usual topological baryon number in the Skyrme model when the same field is interpreted in that theory. With a compatible orientation convention it can be writtenConjugation from the left current to the right current preserves the trace in this formula. Higher dimensions are governed by their corresponding homotopy groups, which need not all be trivial; no assumption of their triviality is needed in the scaling obstruction below.

For the pure sigma model, $E(L)=L^{d-2}{E}_2$, so stationarity requires $(d-2)E_2=0$. In every $d\ne 2$ a finite-energy nonconstant static solution is excluded by Derrick scaling. In $d=1$ spreading lowers its energy; in $d>2$ shrinking lowers it. In particular, a degree-carrying three-dimensional sigma configuration can collapse toward zero size and energy within its sector, with a singular limiting field. The topological charge prevents smooth unwinding, but not this singular collapse.

In $d=2$, the sigma energy is scale invariant. The scaling identity supplies no obstruction, but ${\pi }_2(S^3)=0$ gives no topological soliton sector either. One must not infer nonexistence of every nonconstant stationary harmonic map from these two facts: scale-invariant sigma models can have non-topological harmonic-map saddles, for example using a totally geodesic two-sphere in the target three-sphere. The conclusion is absence of a topologically protected sigma soliton here, with no positive scale-restoring force supplied by the quadratic term alone.

For the Skyrme model, the added commutator term gives the nonnegative static energyAnti-Hermiticity of the commutators fixes its sign. The target and boundary condition are unchanged, so the topological classification is unchanged. The Derrick dimension test for the Skyrme model instead becomesIn $d=3$, the opposite exponents permit a balance:For an arbitrary reference shape with both energies nonzero, its preferred scale is $L_{\ast }=(E_4/E_2{)}^{1/2}$ and its optimized energy is $2\sqrt{E_2{E}_4}$. Expansion costs quadratic-gradient energy and contraction costs quartic-gradient energy. Integer-degree Skyrmions are therefore topologically possible and evade the pure sigma collapse. This mechanism permits finite-size stable solitons, although existence and stability in all field directions require more than the scaling argument itself.

The other dimensions behave differently. In $d=1$, all spatial commutators vanish identically, so the added term changes nothing and nonconstant static finite-energy fields remain excluded. In $d=2$, stationarity forces $E_4=0$; any configuration with nonzero commutator energy can lower it by spreading. In fact a smooth finite-energy static field of this particular potential-free model must be constant. To prove this commuting-current obstruction in a two-dimensional Skyrme model, $E_4=0$ forces $[R_1,R_2]=0$ pointwise. The quartic first variation then vanishes, leaving the sigma equation ${\partial }_i{R}_i=0$. The right-current Maurer-Cartan equation isDivergence-free and curl-free currents have componentwise $\Delta R_i=0$. Their components belong to L2 space by finite $E_2$. An entire square-integrable harmonic function on the plane vanishes: the mean-value formula and Cauchy–Schwarz bound its value by $(\pi R^2{)}^{-1/2}$ times its full L2 norm, tending to zero as the disk radius $R$ grows. Thus $R_i=0$, and the boundary condition fixes $U=1_2$. This additional argument uses the vanishing commutator; it does not apply to the pure sigma model's general noncommuting currents.

In $d=4$, the virial relation gives $2E_2=0$, hence a constant field and zero $E_4$. In $d>4$ both coefficients are positive and the same conclusion follows. Only three spatial dimensions allow a nontrivial finite-size balance of these two positive derivative energies. Higher-dimensional topological classes, where present, do not evade this obstruction.

A SU(2) matrix has the form $\left(\begin{array}{cc}a& b\\ -\overline{b}& \overline{a}\end{array}\right)$ with $|a{|}^2+|b{|}^2=1$. Comparing this with the Pauli expansion givesConsequently all four Pauli coefficients must be real, and their norm condition isConversely, the Pauli matrix multiplication law gives $A{A}^{†}=(u_0^2+|\mathbf{u}{|}^2)I$ and $\det A=u_0^2+|\mathbf{u}{|}^2$, so these conditions imply unitarity and determinant one. For complex coefficients the determinant equation alone would not imply unitarity.

On either open hemisphere,Both signs are necessary globally. The two hemispheres meet along $u_0=0$, and the full four-coefficient constraint is exactly the unit sphere in ${\mathbb{R}}^4$. The coefficient map and its inverse are smooth, establishing SU(2) as the three-sphere as a manifold, not merely a dimension count.

For two real coefficient quadruples, expanding the matrix product with the Pauli matrix multiplication law givesFor an infinitesimal second factor, $v_0=1+O(|\mathbf{v}{|}^2)$. Thereforewhile $d{u}_0=-u_i{v}_i$. These variations are tangent to the unit sphere because $u_{0}d{u}_0+u_{i}d{u}_i=0$.

On a hemisphere, differentiation of the norm constraint gives ${\partial }_{u_i}{u}_0=-u_i/u_0$, and henceContracting with the Pauli-coordinate left-invariant vector fields on SU(2) and using ${\mu }_{ji}{u}_i=u_0{u}_j$ givesThe Pauli matrix multiplication law gives exactly the same expression for $-A{\sigma }_j$, soIt follows thatThis uses the Pauli matrix commutator identity. The commutator of two first-order vector fields is again a vector field, and the matrix entries of $A$ span the four coordinate functions $u_0,u_1,u_2,u_3$. Equality of their actions on these entries determines the tangent field. ThereforeThe real tangent field underneath $T_j$ is generated by the flow $A\mapsto A{e}^{it{\sigma }_j}$; left multiplication transports that tangent consistently, explaining its left invariance. The rescaled generators $-T_j/2$ have the usual positive-sign angular-momentum commutator.

To determine the Haar measure, compute the metric induced from the unit three-sphere. Since $d{u}_0=-u_{i}d{u}_i/u_0$, its line element isThe rank-one matrix determinant lemma givesThus the Pauli-coordinate Haar measure on SU(2) isup to an overall normalization. To check its group invariance, fix one unit coefficient quadruple and multiply by it. The product formulas define a linear transformation of the other quadruple. The norm of a product is the product of norms, as follows either from $A{A}^{†}$ or directly from the dot and cross products. Fixed multiplication by a unit quadruple is therefore an orthogonal transformation of ${\mathbb{R}}^4$. It preserves the round sphere metric and volume, on both the left and the right, proving that this is the invariant measure.

The same density can be checked directly against the vector fields. On a fixed hemisphere, $u_0/|u_0|$ is constant, andThe second term vanishes because its contractions involve either ${\delta }_{ik}$ or $u_i{u}_k$, both symmetric in $i,k$. Thus the density has zero divergence along every invariant tangent direction.

The hemisphere chart becomes singular at the equator, but its volume is finite:Both hemispheres give total volume $2{\pi }^2$, so probability-normalized Haar measure is $d^{3}u/(2{\pi }^2|u_0|)$ on each hemisphere. Finally, the three-sphere is closed and bounded in ${\mathbb{R}}^4$, hence compact; its smooth identification with SU(2) proves that $SU(2)$ is a compact group. The finite integral reflects this compactness, and the equatorial divergence is only a coordinate artefact.

The geometric object underlying an $SU(2)$ gauge field is a principal connection on a principal bundle $P\to M$ with structure group SU(2). The base $M$ is spacetime, or an oriented Riemannian manifold in the Euclidean formulation. A choice of local section identifies each fibre with SU(2) and produces a local gauge potential. Such a choice is a choice of internal frame; changing it produces a gauge equivalence of principal connections. The bundle specifies the global gluing of these frames, while the principal connection specifies how frames at neighbouring points are compared.

The SU(2) group consists of $2\times 2$ unitary matrices of determinant one. Its Lie algebra consists of traceless Skew-Hermitian matrices. For definiteness usewhere the ${\sigma }_a$ are Pauli matrices. The commutator fixes all signs below. The positive invariant inner product on this Lie algebra is $-\mathrm{tr}(XY)$. Absorb the coupling into the gauge potential, so the gauge covariant derivative in the defining two-dimensional representation is $D=d+A$ with $A=A^a{t}_a$. A convention using Hermitian Pauli matrices instead moves factors of $i$ and the coupling into $D$; the underlying principal connection is the same geometric data.

A principal connection can be given by a Lie algebra-valued differential one-form $\mathcal{A}$ on $P$ satisfyingHere ${\xi }_P$ is the vertical vector generated by the right group action. The first identity recovers vertical motion and the second makes the construction independent of a fibre frame. Its kernel is the horizontal distribution of a principal connection, complementary to the vertical tangent spaces. A curve is horizontal when $\mathcal{A}$ annihilates its velocity, so a principal connection defines parallel transport along curves in $M$.

For a local section $s$, the local principal connection form is $A=s^{\ast }\mathcal{A}$. Change the section to $s^{\prime }=sh$, with $h:U\to SU(2)$. Differentiating $sh$ has a translated horizontal part and a vertical part from $dh$. The two defining identities for the principal connection therefore giveOn overlapping trivializations this is exactly the compatibility law for the local principal connection forms. In particular, the gauge potentials need not glue as ordinary globally defined differential one-forms on $M$. On a nontrivial principal bundle there need not be any global section with respect to which one could write a single matrix-valued $A$.

An active gauge equivalence of principal connections is generated by a bundle automorphism covering the identity on $M$ and commuting with the right SU(2) action. Its local description has the same transformation formula, but its local group-valued functions must satisfy transition compatibility on overlaps. This distinguishes the structure group SU(2) from the full gauge group of a fixed bundle. Two descriptions related by a passive change of section encode the same principal connection; two connections related by such an automorphism lie in the same physical gauge orbit. The space of principal connections on a fixed bundle is affine: the difference of two connection forms is a differential one-form with values in the adjoint bundle, since their inhomogeneous transformation terms cancel.

The curvature of a principal connection isIt is horizontal and equivariant, hence descends to a differential two-form on $M$ valued in the adjoint bundle. In a local section the gauge curvature becomesThe product $A\wedge A$ combines the exterior product with matrix multiplication. The commutator term is the specifically non-Abelian interaction. Substitution of the transformation of $A$, using $d(h^{-1})=-h^{-1}(dh)h^{-1}$, cancels all derivatives of $h$ and givesThus the gauge curvature transforms tensorially even though the gauge potential does not. Geometrically, the curvature of a principal connection measures the failure of the horizontal distribution of a principal connection to be integrable; physically it is the non-Abelian gauge field strength.

Matter fields are sections of associated vector bundles. A field in the defining representation of SU(2) is a section of $E=P{\times }_{SU(2)}{\mathbb{C}}^2$. Its two local components satisfy ${\psi }^{\prime }=h^{-1}\psi$ under the section convention above. The induced gauge covariant derivative obeysThis covariance is why replacing ordinary derivatives by gauge covariant derivatives makes the kinetic terms compatible with changes of internal frame. Applying the gauge covariant derivative twice yields $D^2\psi =F\psi$, so the gauge curvature also measures the noncommutativity of covariant differentiation. Other group representations give other associated vector bundles and replace $A$ by its image under the differential of the representation.

For an adjoint-valued differential form $\Phi$ of degree $r$, setExpansion of $F=dA+A\wedge A$, using $d^2=0$, proves the gauge-theory Bianchi identityThis is an identity for every principal connection, independent of any field equation. It should be distinguished from the dynamical Yang-Mills equations.

To obtain dynamics, equip $M$ with a Riemannian metric and an orientation, and let $\ast$ be its Hodge star operator. With the Skew-Hermitian normalization above, the positive Euclidean Yang-Mills action isThe matrix trace and the Hodge star operator make this independent of the choice of local section. The minus sign compensates for the negative trace form on Skew-Hermitian matrices. Varying the principal connection gives $\delta F=D_A\delta A$, and thereforeFor compactly supported variations, or on a closed base, integration by parts gives the Yang-Mills equationsIn components these are $D_{\mu }{F}^{\mu \nu }=0$. The gauge-theory Bianchi identity provides the complementary geometric identity $D_{A}F=0$. Coupling matter adds the associated gauge current to the dynamical equation; the transformation law of the gauge covariant derivative ensures covariance of that current as well.

The principal connection also supplies nonlocal observables. Along a path $\gamma$, parallel transport is determined byThe ordering is necessary because the Lie algebra matrices at different points need not commute. The resulting Wilson line transforms as $U_{\gamma }^{\prime }=h(\gamma (1){)}^{-1}{U}_{\gamma }h(\gamma (0))$. For a closed path, its matrix trace is a Wilson loop, independent of the frame at the basepoint. These observables record the holonomy of a connection. A flat principal connection has trivial holonomy of a connection around sufficiently small contractible loops but may have nontrivial holonomy of a connection around noncontractible loops. Thus vanishing local gauge curvature need not eliminate global gauge information.

The global topology becomes particularly visible in four dimensions. For a closed oriented four-dimensional base, Chern-Weil theory gives the Second Chern number of the defining associated vector bundle:The plus sign here uses Skew-Hermitian curvature and the mathematical second-Chern convention: expanding $\det (I+iF/(2\pi ))$ with $\mathrm{tr}F=0$ gives precisely the displayed Second Chern form. A convention defining a physics topological charge with a leading minus sign instead calls that charge $-k$. Neither choice changes the absolute-value action bound.

The Second Chern number is independent of the principal connection on a fixed bundle. Indeed, the gauge-theory Bianchi identity implieswhose integral vanishes on a closed base by Stokes theorem. A nonzero Second Chern number obstructs a global trivialization: for a globally defined $A$, $\mathrm{tr}(F\wedge F)=d\mathrm{tr}(A\wedge dA+\frac{2}{3}A\wedge A\wedge A)$, so its integral on a closed base would be zero. On $S^4$, gluing two trivial bundles over four-balls uses a transition map $S^3\to SU(2)$. Since SU(2) as the three-sphere identifies the target with $S^3$, such gluing maps have integer degree of a map between oriented manifolds, illustrating how distinct topological sectors arise.

In four Euclidean dimensions, the Hodge star operator squares to one on differential two-forms. Decompose $F=F_{+}+F_{-}$ with $\ast F_{\pm }=\pm F_{\pm }$. Define the nonnegative squared norms $a_{\pm }=-\int \mathrm{tr}(F_{\pm }\wedge \ast F_{\pm })$. Orthogonality givesEquality means that one component vanishes, giving the self-dual Yang-Mills equations or the Anti-self-dual Yang-Mills equations. Such finite-action solutions are Yang-Mills instantons. They solve the full Yang-Mills equations because $D_A\ast F=\pm D_{A}F=0$. With the second-Chern convention just specified, anti-self-dual curvature has $k\ge 0$ and self-dual curvature has $k\le 0$. On noncompact spacetime one must impose suitable decay or boundary conditions before treating the topological integral as an integer and using the same bound without boundary terms.

Local gauge potentials, matter gauge covariant derivatives, gauge field strength, parallel transport and Second Chern number are different manifestations of one principal connection. The principal bundle preserves the global information that a single local gauge potential cannot capture; the Yang-Mills action then selects its physical dynamics.

A Lie group is a finite-dimensional smooth manifold with a group structure for which multiplication and inversion are smooth. The group defined here is the compact symplectic group $Sp(n)=U(2n)\cap Sp(2n,\mathbb{C})$, rather than the full complex symplectic group. Also, the displayed matrix expression requires $B$ to be $2n\times 2n$; the printed $n\times n$ size is incompatible with $J$.

Since $J^2=-I$, we have $J^{-1}=-J$. The matrix exponential commutes with conjugation, as follows term by term from its absolutely convergent power series. ConsequentlyTo construct logarithm charts for the compact symplectic group, consider the real vector spaceDifferentiating the defining identities at $I$ gives precisely these conditions. Conversely if $X\in \mathfrak{k}$, $e^X$ is unitary andso $e^X\in Sp(n)$. These are the infinitesimal conditions of the compact symplectic Lie algebra.

Near $I$, the convergent matrix logarithm series $\log (I+C)={\sum }_{m\ge 1}(-1{)}^{m+1}{C}^m/m$ is smooth and inverse to the matrix exponential near zero. These local inverses respect transpose, conjugate transpose, and conjugation; also $\log (A^{-1})=-\log A$ when both matrices are sufficiently close to $I$. Shrink their neighborhoods accordingly. If $A\in Sp(n)$ there, unitarity givesThe symplectic identity is equivalent to $A^t=J^{-1}{A}^{-1}J$, so, putting $X=\log A$, it gives $X^t=-J^{-1}XJ=JXJ$, equivalently $XJ+J{X}^t=0$. Thus this local logarithm restricts to a bijection between a neighborhood of $I$ in $Sp(n)$ and an open neighborhood of zero in $\mathfrak{k}$. Its inverse is the restricted matrix exponential. These restrictions are manifold charts; left multiplication translates them to every $A_0\in Sp(n)$ via $A\mapsto \log (A_0^{-1}A)$. The chart overlaps are smooth compositions of multiplication, exponential and logarithm. The subspace topology is Hausdorff and second countable, inherited from the finite-dimensional matrix space, so these charts give a smooth manifold.

Closure under products and inverses follows from $AJ{A}^t=J$ and unitarity. Matrix multiplication is polynomial in real and imaginary entries, and inversion on the unitary group is $A\mapsto A^{\ast }$, a real linear operation. Their restrictions are smooth in the charts just constructed. Hence $Sp(n)$ is a Lie group without needing a closed-subgroup theorem.

Write $X$ in $n\times n$ blocks. The two infinitesimal conditions giveThe skew-Hermitian matrix $P$ has $n^2$ real parameters; the complex symmetric matrix $Q$ has $n(n+1)/2$ complex parameters, hence $n(n+1)$ real parameters. ThereforeFor $n=1$, any two-by-two matrix satisfies $AJ{A}^t=(\det A)J$, so the group is $SU(2)$. Explicitly,The rows are orthonormal and the determinant is one; conversely unitarity and determinant one force this form. This is the SU(2) as the three-sphere parametrization. The map and its inverse, extraction of the first row, are smooth. Thus $Sp(1)$ is diffeomorphic to $S^3$.

Rescale the ambient coordinates by $q_i=a_i{r}_i$. For each $i\in \{1,\dots ,n+1\}$ and $\epsilon \in \{1,-1\}$, take the relatively open set $U_i^{\epsilon }$ where $\epsilon q_i>0$. Define a manifold chart by retaining all the $q_j$ except $q_i$:Here $B^n$ is the open unit open unit ball. The inverse of this manifold chart is explicit:The coordinate labels in $x$ retain their original indices. All the $a_i$ are nonzero, so this inverse exists; the strictly positive radicand makes it smooth on the whole open open unit ball. Projection and this inverse are continuous, so each manifold chart is a homeomorphism onto an open subset of ${\mathbb{R}}^n$.

These $2(n+1)$ manifold charts cover the ellipsoid, because at every point at least one $q_i$ is nonzero. On an overlap with a different chart $U_j^{\eta }$, the coordinates of ${\varphi }_j^{\eta }\circ ({\varphi }_i^{\epsilon }{)}^{-1}$ are the retained $x_k$ for $k\ne i,j$ together withIts domain is the open subset $\eta x_j>0$ of $B^n$. The displayed functions, and the reverse transition functions obtained by exchanging $i,j$, are smooth. Transitions between identical charts are the identity; opposite-sign charts with the same omitted coordinate do not overlap. The ambient subspace topology is Hausdorff and second countable, as it is inherited from Euclidean space. Thus this coordinate-projection atlas of an ellipsoid is a smooth atlas, proving the ellipsoid is a smooth manifold of dimension $n$. The rescaling $r\mapsto q$ also gives a diffeomorphism with the unit sphere $S^n$.

For the three-sphere, use its identification with SU(2) rather than separate local constructions. With the Pauli matrices ${\tau }_i$, put $T_i=-i{\tau }_i$. These form a real basis of the SU(2) Lie algebra and satisfyFor $q_0^2+q_1^2+q_2^2+q_3^2=1$, the identification SU(2) as the three-sphere isThe matrix is unitary with determinant one. Conversely, every SU(2) matrix has this form, and both the correspondence and its inverse are smooth.

The left translations $L_g(h)=gh$ are diffeomorphisms. Define three left-invariant vector fields byMatrix multiplication makes these vector fields smooth globally. Since $(d{L}_g{)}_e$ is an isomorphism from the SU(2) Lie algebra to $T_g\mathrm{SU}(2)$, the three vectors are linearly independent at every point, and each is nowhere zero. This is the parallelization of a Lie group by left translations.

Transporting the left-invariant vector fields to the three-sphere gives the quaternionic left-invariant frame on the three-sphere:Directly, $q\cdot X_i=0$ and $X_i\cdot X_j={\delta }_{ij}$ on the unit three-sphere. Thus the vectors are tangent and form a global orthonormal frame of a vector bundle. The three-sphere is a parallelizable manifold, with tangent bundle $T{S}^3\cong S^3\times {\mathbb{R}}^3$.

The two Lie groups have the same local infinitesimal structure, but different global topology. This difference determines which Lie algebra representations integrate to representations of each group.

An element of the SU(2) group is a unitary matrix of determinant one. Orthogonality of its columns and its determinant give the unique formThus its group manifold is the unit three-sphere in ${\mathbb{C}}^2\simeq {\mathbb{R}}^4$: this is SU(2) as the three-sphere. In particular it is compact, connected and simply connected. In terms of the Pauli matrices one may also write $U=a_{0}I-i\mathbf{a}\cdot \mathbf{\sigma }$ with real coefficients satisfying $a_0^2+{\mathbf{a}}^2=1$.

Differentiate $U(t{)}^{†}U(t)=I$ and $\det U(t)=1$ at $U(0)=I$. The tangent space consists of traceless skew-Hermitian matrices:The Lie bracket of a Matrix Lie group is the matrix commutator. With $T_a=-i{\sigma }_a/2$, the Pauli matrix multiplication law givesThis derives the SU(2) Lie algebra as a three-dimensional real Lie algebra. The Hermitian physics generators $J_a={\sigma }_a/2$ instead obey $[J_a,J_b]=i{ϵ}_{abc}{J}_c$; they are $i$ times the skew-Hermitian tangent generators, so these are consistent conventions.

The SO(3) group consists of real orthogonal matrices with determinant one. Differentiating $R(t{)}^{T}R(t)=I$ at the identity givesThe determinant condition gives no additional infinitesimal constraint because a skew-symmetric matrix already has trace zero. Define $L_a\mathbf{v}={\mathbf{e}}_a\times \mathbf{v}$. The cross product identity impliesTherefore the SO(3) Lie algebra has the same structure constants and $T_a\mapsto L_a$ is a Lie algebra isomorphism.

The global relation is the Adjoint double cover from SU(2) to SO(3). For $V=\mathbf{v}\cdot \mathbf{\sigma }$, define $R(U)$ byConjugation preserves the real space of traceless Hermitian matrices and its inner product $\frac{1}{2}\mathrm{tr}(VW)=\mathbf{v}\cdot \mathbf{w}$. Hence $R(U)$ is orthogonal. Continuity and connectedness, together with $R(I)=I$, put it in $SO(3)$. Composition of conjugations makes $R$ a group homomorphism. If $R(U)=I$, then $U$ commutes with every Pauli matrix, hence is scalar; unitarity and determinant one leave precisely $U=\pm I$. The differential sends $T_a$ to $L_a$, so it is an isomorphism. More concretely,induces rotation through angle $\theta$ about $\mathbf{n}$, by the Rodrigues rotation formula. Every three-dimensional rotation has such an axis and angle, proving surjectivity. ConsequentlyThe matrices $U$ and $-U$ are antipodal points on the three-sphere, so the SO(3) group manifold is Real projective space ${\mathbb{RP}}^3$. Equivalently, the closed axis-angle ball $\Vert \theta \mathbf{n}\Vert \le \pi$ has opposite boundary points identified. The fundamental group is ${\pi }_1(SO(3))\simeq {\mathbb{Z}}_2$, whereas ${\pi }_1(SU(2))=0$. A $2\pi$ rotation lifts from $I$ to $-I$; a $4\pi$ rotation returns to $I$. Thus the covering is the universal cover and the groups are not globally isomorphic.

For representation theory, specify finite-dimensional complex continuous representations. Compactness permits an invariant Hermitian inner product, obtained by averaging against Haar measure, and therefore complete reducibility. Complexifying either real Lie algebra gives the sl2 Lie algebra. Its finite-dimensional irreducibles are indexed by $n\in {\mathbb{Z}}_{\ge 0}$, have highest weight $n$, and have dimension $n+1$. By integration of a Lie-algebra representation, since $SU(2)$ is simply connected, every such Lie algebra representation integrates uniquely. The resulting homogeneous polynomial representation of SU2 isIn the spin angular momentum notation $j=n/2$, its Hermitian $J_3$ eigenvalues are $j,j-1,\dots ,-j$. The central matrix $-I$ acts on the symmetric power by $(-1{)}^n$, so descent of an SU(2) representation to SO(3) occurs exactly when $n$ is even. HenceFor a reducible representation, every summand must satisfy the descent condition. The spin-one-half doublet is a genuine representation of the covering group but does not define a single-valued representation of $SO(3)$; the spin-one triplet does and is its vector representation. The distinction is topological, rather than a difference in their isomorphic Lie algebras.

Let $N=(0,\dots ,0,1)$ and $S=(0,\dots ,0,-1)$ be the poles of $S^n$. The open sets $U_N=S^n\setminus \{N\}$ and $U_S=S^n\setminus \{S\}$ cover the sphere. Stereographic projection gives coordinate chartsfrom these sets to ${\mathbb{R}}^n$. Their inverses areOn the overlap, the transition map iswhich is a smooth diffeomorphism of ${\mathbb{R}}^n\setminus \{0\}$. These two compatible charts make $S^n$ a smooth manifold of dimension $n$.