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A linear representation $\rho :G\to GL(V)$ of a principal bundle's structure group defines the quotient of $P\times V$ by $(p,v)\sim (pg,\rho (g{)}^{-1}v)$. It is a vector bundle over the same base. A principal connection induces parallel transport and a local covariant derivative $d+{\rho }_{\ast }(A)$ on its sections.

A connection on a vector bundle pulls back along a smooth curve $\gamma$ to differentiate its time-dependent sections. For the tangent bundle, a section $V(t)$ of the pullback tangent bundle hasThe Leibniz rule under a change of frame proves independence of the coordinates. The derivative includes the time dependence of the coefficients, so it is meaningful even at a zero tangent velocity. A vector field along a map need not extend to one ambient field if the curve self-intersects. Parallel fields satisfy $D_{t}V=0$, a linear ordinary differential equation whose unique solutions define parallel transport.

Transport defining a nonrotating frame along an accelerated timelike curve. With four-velocity normalization $u^{\mu }{u}_{\mu }=-c^2$ and four-acceleration $a^{\mu }=D{u}^{\mu }/d\tau$, the displayed equation preserves inner products and keeps a spatial spin orthogonal to $u$. Along a geodesic, $a^{\mu }=0$ and it reduces to parallel transport.

A gyroscope is a rotating instrument whose rapidly spinning, nearly symmetric rotor provides a direction reference through its angular momentum. In an inertial frame, negligible external torque approximately preserves that direction by conservation of angular momentum. A transverse torque turns the angular momentum instead: the surprising sideways motion of a supported spinning wheel is precession. An ideal freely falling test gyroscope in general relativity has its spin undergo parallel transport along its timelike geodesic. Comparing that direction with distant reference axes measures Lense-Thirring precession; the direction need not remain fixed in arbitrary rotating coordinates.

A gyroscope direction reference is useful precisely because its direction can persist while its housing turns. A spinning-wheel precession experiment deliberately supplies gravitational torque to reveal how that persistence fails. The gyroscope vector geometry explains the direction, and slow steady gyroscope precession gives a numerical prediction with a check on its assumptions.

For the Riemannian metric $d{r}^2+f(r{)}^{2}d{\varphi }^2$, where $f>0$ and $\varphi$ has period $2\pi$, parallel transport around $r=r_0$ acts on the orthonormal basis components of a tangent vector by a rotation through $-2\pi f^{\prime }(r_0)$. Indeed, those components obey $u^{\prime }=f^{\prime }(r_0)v$ and $v^{\prime }=-f^{\prime }(r_0)u$ as functions of $\varphi$. Thus all tangent vectors return to themselves exactly when $f^{\prime }(r_0)\in \mathbb{Z}$. The integer criterion concerns holonomy and is stronger than local flatness.

On an oriented even-dimensional Riemannian manifold of strictly positive sectional curvature, a nonconstant closed geodesic has a length-decreasing smooth variation. Its parallel transport fixes the tangent and acts on the odd-dimensional normal space by a special orthogonal group element. An odd-dimensional special orthogonal transformation has a fixed vector, so there is a nonzero periodic parallel normal field $V$. The second variation of geodesic energy is then strictly negative. The Cauchy-Schwarz inequality converts lower energy into strictly lower length because the original geodesic has constant speed. For an embedded geodesic the small variation is a smooth isotopy; otherwise it is a deformation through immersed loops.

A Killing-Yano tensor is a differential form satisfying the displayed equation for the Levi-Civita connection. Its covariant derivative is totally antisymmetric. Contraction with the velocity of an affinely parametrized geodesic gives a form that is carried by parallel transport along that geodesic. This differs from a Killing tensor, which is symmetric.

For a Killing-Yano two-form, ${\nabla }_a{Y}_{bc}$ is a differential three-form. Along a geodesic with velocity $v$, the covector $w_c=Y_{bc}{v}^b$ is carried by parallel transport, since $v^a{v}^b{\nabla }_a{Y}_{bc}=0$.

Author's graduate lecture notes, arXiv:gr-qc/9712019, 1997. Section 2 develops tangent vectors, covectors and the failure of ordinary tensor derivatives under nonlinear coordinate changes; section 3 constructs covariant differentiation, parallel transport and curvature. Metric and curvature signs must be translated when combining conventions.

In a local trivialization, a vector-bundle connection with matrix $A$ defines the displayed complement of the vertical tangent space. The frame-change law makes these complements independent of the trivialization. Their dependence on the fiber coordinate is linear; an arbitrary horizontal complement need not satisfy this condition and need not define a connection on a vector bundle. Horizontal lifts satisfy the ordinary differential equation defining parallel transport.

A Riemannian manifold is locally symmetric when its Riemann curvature tensor is parallel for the Levi-Civita connection. Along a geodesic, parallel transport then conjugates the Jacobi curvature operator at the initial point to the operator at every later point. Thus its eigenvalues stay constant. The condition is local and does not assert that global point reflections extend to isometries of the whole manifold.

The Möbius band is the topological surface obtained by gluing the ends of a rectangle with a half-turn. Precisely, it is the quotient spaceThe twist identifies the two apparent long edges into one boundary circle. Following the core circle of a Möbius band reverses local handedness, so this surface is not orientable. An untwisted untwisted band has two boundary circles and is orientable: the distinction survives stretching and bending because orientability is a topological invariant.

To explore the construction with paper, mark the two long edges in different colours, join the short ends after a half-turn, and trace an edge. The trace passes from one marked edge to the other without ever leaving the boundary. Separately, trace the centre line and track a local transverse arrow: it comes back reversed after one circuit. The boundary closes after twice the travel around the centre, while the transverse arrow needs two circuits to return to its original direction.

This original mathematical model has central radius $R=2$ and half-width $a=0.55$:Its seam obeys $F(2\pi ,w)=F(0,-w)$. The dotted circle is the core circle of a Möbius band; the teal curve follows the single boundary through $0\le \theta \le 4\pi$. At the same point on the core, the arrows for $\theta =0$ and $\theta =2\pi$ are opposite. The graph compares a continuously followed unit normal with its starting value in the ambient space; the endpoint reversal, rather than any intermediate zero of the graph, establishes the obstruction. It is not a statement about parallel transport.

The Möbius band has the homotopy type of a circle: a deformation retraction shrinks each width coordinate to zero. Nevertheless its boundary represents twice a generator of its fundamental group. The boundary of a Möbius band winds twice around its core, so with compatible generator choices the boundary generator maps to $c^2$ in ${\pi }_1(M)\cong \mathbb{Z}$.

The historical account by Julyan H. E. Cartwright and Diego L. González places the mathematical discovery by August Möbius and Johann Listing in 1858, while investigating representations of twisted bands predating that work. A familiar-looking picture and a recognition of the underlying topological invariant are different kinds of evidence. ^[ref]

For an affine null geodesic congruence with future tangent $U$, choose a future unit timelike vector $T$ on a transverse section, put $E=-U\cdot T$, and set $N=T/E-U/(2E^2)$ there. It has $N^2=0$ and $U\cdot N=-1$. Parallel transport along each generator preserves these contractions by metric compatibility and ${\nabla }_{U}U=0$. The construction is local up to caustics and supplies the auxiliary normal for a screen-space projector.

A differential form is parallel when its covariant derivative vanishes. Parallel transport then preserves it, and every parallel form is harmonic.

The form is preserved by parallel transport; this is the alternating-tensor case of vanishing covariant derivative.

The Levi-Civita connection of an oriented Riemannian manifold preserves its Riemannian volume form. In a positively oriented orthonormal frame its connection matrix is skew-symmetric. Differentiating the wedge of the dual coframe gives minus the trace of that matrix times the volume form, which is zero. Equivalently parallel transport preserves the metric and orientation, so it preserves the normalized volume form.

Two Riemannian metrics have the same Levi-Civita connection precisely when their relative positive self-adjoint endomorphism $A$ is parallel for that connection. Parallel transport preserves both metrics. Consequently agreement at one point implies agreement everywhere on a connected manifold. Without that agreement, constant scalar multiples always give examples; constant nonproportional Euclidean forms show that a common connection does not force scalar proportionality.

A parallel spinor field is annihilated by the spinor covariant derivative. Its value at one point determines it on a connected domain by parallel transport. The spinor curvature identity implies $R_{\mu \nu AB}{\gamma }^{AB}ϵ=0$, but this local condition alone need not settle global existence: the spin structure and holonomy also matter. Lorentzian parallel spinors need not force the whole Ricci tensor to vanish, unlike the positive-definite case.

For the positively oriented boundary of a spherical triangle on the unit sphere, parallel transport is rotation by its angle excess modulo $2\pi$. The tangent is parallel along each geodesic side and turns by $\pi -\alpha$ at a corner of angle $\alpha$. The three corner turns sum to $3\pi -\alpha -\beta -\gamma$, and combining them with parallel transport returns the initial tangent. Hence the transport angle is the negative of that sum modulo $2\pi$, giving the formula. The spherical excess formula identifies the angle with enclosed area; reversing traversal changes its sign.

For a metric connection on a vector bundle, the covariant derivative along a curve satisfies $(d/dt)⟨u,v⟩=⟨D_{t}u,v⟩+⟨u,D_{t}v⟩$. If both sections are parallel, the right side vanishes. Thus parallel transport preserves the fibre inner product, not merely vector lengths, and is a linear isometry between the endpoint fibres.

A vector field is parallel for a specified connection on a vector bundle on the tangent bundle when its covariant derivative in every direction vanishes. Its values along every smooth curve are related by parallel transport. For a Levi-Civita connection, its length is constant on each connected component, since the connection preserves the Riemannian metric.