Exponential map (Riemannian geometry, ${\exp }_p$)

For $p$ in a Riemannian manifold and $v$ sufficiently close to zero in $T_{p}M$, let ${\gamma }_v$ be the geodesic with ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. The exponential map isEquivalently, ${\exp }_p(tv)={\gamma }_v(t)$ wherever both sides are defined.

For the Riemannian exponential map ${\exp }_p(v)={\gamma }_v(1)$, uniqueness of the geodesic equation gives ${\exp }_p(tv)={\gamma }_v(t)$. Differentiation at $t=0$ yields $(d{\exp }_p{)}_{0}v=v$. Smooth ODE dependence and the inverse function theorem therefore make ${\exp }_p$ a diffeomorphism on a neighborhood of zero, producing geodesic normal coordinates. This is a local assertion and requires no geodesic completeness.

A convex normal neighbourhood is an open set whose points have unique smoothly varying joining geodesics, with the joining geodesic contained in the set. The corresponding star-shaped restriction of each exponential map is a diffeomorphism onto the neighbourhood. Hence its differential is invertible and no such joining geodesic admits a nonzero Jacobi field vanishing at both endpoints. Every point of a Riemannian manifold has arbitrarily small such neighbourhoods. This is stronger than merely requiring the existence of a minimizing geodesic in the set.

The geodesic flow sends a tangent vector $v$ to the velocity at time $t$ of the unique geodesic with initial velocity $v$. Smooth dependence of solutions of the geodesic equation makes it a smooth flow on the tangent bundle wherever defined.

A perpendicular Jacobi field on a Riemannian surface has scalar equation $\ddot{y}+Ky=0$. For strictly negative curvature on a closed manifold, the displayed derivative is uniformly positive definite, so the quadratic-form criterion for an Anosov flow makes the geodesic flow Anosov. When $K=-1$, the stable and unstable lines in the canonical frame are respectively $\mathbb{R}(H-V)$ and $\mathbb{R}(H+V)$, with factors $e^{-t}$ and $e^t$.

A smooth real function $\lambda$ on the unit tangent bundle determines this unit-speed smooth flow. Its base velocity is $v$, so its generator never vanishes. In the canonical frame its divergence in Liouville volume of a surface geodesic flow is $V\lambda$. The special case where $\lambda$ depends only on the base is a magnetic flow; when $\lambda (x,v)=⟨E(x),iv⟩$ it is a Gaussian thermostat.

Write a tangent variation of a surface thermostat as $aF+yH+zV$ in the canonical frame. The commutators $[F,H]=-\lambda F+(K-H\lambda +{\lambda }^2)V$ and $[F,V]=-H-(V\lambda )V$ give $\dot{a}=\lambda y$ and the stated transverse equation. Near the vertical projective direction, $r=y/z$ obeys $\dot{r}=1-(V\lambda )r+(K-H\lambda +{\lambda }^2)r^2$.

At unit speed on a Riemannian surface, the right-hand side is $⟨E,iv⟩iv$. This removes the velocity-parallel component of the forcing, so differentiating $|\dot{\gamma }{|}^2$ gives zero. The dynamics need not preserve the ordinary Liouville volume of a surface geodesic flow.

For $F=X+\lambda V$, the canonical fields $X,V$ preserve Liouville volume of a surface geodesic flow, so the divergence is $V\lambda$. For a Gaussian thermostat, $\lambda =⟨E,iv⟩$ and $V(iv)=-v$, giving the stated sign. If $E=\nabla f$, then the density $e^{f\circ \pi }$ makes this divergence zero.

A unit-speed curve with $D_t\dot{\gamma }=f(\gamma )i\dot{\gamma }$ defines the magnetic flow. The vertical term rotates the velocity while the horizontal part transports it along the base. Since $V(f\circ \pi )=0$, the divergence of $X+fV$ in Liouville volume vanishes. The associated two-form on the unit tangent bundle is $d\alpha -f{\pi }^{\ast }{\Omega }_a=-{\iota }_F\mu$, whose kernel is the magnetic flow direction.

On a connected closed oriented surface, a magnetic Anosov flow with $\int f{\Omega }_a=0$ preserves a smooth contact form only when $f=0$. To see the obstruction, take $d\eta =f{\Omega }_a$ and $\sigma =\alpha -{\pi }^{\ast }\eta$. An invariant contact form can be normalized to evaluate to one on $F$; its differential is a constant multiple of ${\iota }_F\mu$, since smooth invariant functions are constant. Averaging closed one-forms forces that multiple to be $-1$, so the contact form differs from $\sigma$ by a closed form. Its fibre-average decomposition gives $Fh=(\eta -\xi )(v)-af$. The degree-zero/one kernel theorem gives $af=0$ and $d\eta =d\xi =aK{\Omega }_a$, forcing $f=0$ whether $a$ vanishes or not. For $f=0$, the canonical form $\alpha$ is invariant and contact.

The metric defines this globally smooth Hamiltonian on the cotangent bundle. With $\omega =d{x}^i\wedge d{p}_i$ and ${\iota }_{X_H}\omega =dH$, its Hamiltonian vector field projects to affinely parametrized geodesics. The momentum is $p_i=g_{ij}{\dot{x}}^j$, and its conserved value is half the squared speed.

The domain of ${\exp }_p$ is the star-shaped open setSmooth dependence for ordinary differential equations makes $(p,v)\mapsto {\exp }_p(v)$ smooth on its open domain in $TM$.

For $S=({\mathbb{R}}^2\setminus \{0\})\times \{0\}\subset {\mathbb{R}}^3$ and $p=(1,0,0)$, geodesics are straight lines until they hit the missing origin. Thus $(-2,0,0)\notin {\mathcal{D}}_p$, and $(-1,0,0)$ is not in the image of ${\exp }_p$ because its unique straight segment from $p$ passes through the origin.

A normal neighbourhood of $p$ is the diffeomorphic image under ${\exp }_p$ of a star-shaped neighbourhood of $0$ in $T_{p}M$. In a sufficiently small normal ball, radial geodesics from $p$ uniquely minimize length.

In a normal neighbourhood $U={\exp }_p(V)$ with $V$ star-shaped, define $r(q)=|{\exp }_p^{-1}(q)|$. Away from $p$, the Gauss lemma gives $\nabla r=(d{\exp }_p{)}_v(v/|v|)$ for $v={\exp }_p^{-1}(q)$. Thus $|\nabla r|=1$ and this unit vector field is tangent to unit-speed radial geodesics. The function $r$ is not smooth at $p$, but $r^2/2$ is smooth and $\nabla (r^2/2)=(d{\exp }_p{)}_{v}v$ extends by zero. On a sufficiently small normal ball, $r$ equals the Riemannian distance from $p$.

// Target: riemannian-geometry.bigb

The unit radial vector field is $E=\nabla r$ on the punctured normal neighbourhood, where $r$ is the radial distance in a normal neighbourhood. Along ${\gamma }_u(t)={\exp }_p(tu)$ with $|u|=1$, it satisfies $E({\gamma }_u(t))={\dot{\gamma }}_u(t)$. The scaled radial field $W=rE=\nabla (r^2/2)$ is smooth at the centre and vanishes there. These conventions have different associated radial functions.

// Target: riemannian-geometry.bigb

The injectivity radius at $p$ is the largest radius on which ${\exp }_p$ is a diffeomorphism from the tangent-space ball onto its image. The injectivity radius of a compact Riemannian manifold is the positive infimum of these pointwise radii.

Choose an orthonormal basis of $T_{p}M$ and use its linear coordinates on a neighborhood of zero on which ${\exp }_p$ is a diffeomorphism. Transporting these coordinates through ${\exp }_p$ gives geodesic normal coordinates centered at $p$.

In geodesic normal coordinates at $p$, radial geodesics have coordinates $x(t)=tv$. The geodesic equation at $t=0$ gives ${\Gamma }_{jk}^i(p)v^j{v}^k=0$ for every $v$. The Levi-Civita connection is torsion free, making these coefficients symmetric in $j,k$; the polarization identity then gives ${\Gamma }_{jk}^i(p)=0$. This does not imply vanishing curvature, which depends on derivatives of the coefficients.

A geodesic sphere is a level set of Riemannian distance from a point. For sufficiently small positive $r$, it is the exponential map image of the radius-$r$ tangent sphere, a smooth compact hypersurface. The Gauss lemma makes its tangent space orthogonal to the radial direction. Larger distance spheres need not be smooth at the cut locus.

A chart $x$ centered at $p$ on a star-shaped coordinate domain is a geodesic normal chart exactly when $g_{ij}(0)={\delta }_{ij}$ andfor every $x$ and $k$. This is the geodesic equation for every radial curve $t\mapsto tx$.

Choose an orthonormal basis of $T_{p}M$ and put $e(\theta )=(\cos \theta ,\sin \theta )$. Away from the centre and cut locus,defines geodesic polar coordinates centred at $p$.

Before the cut locus, define $A(X)={\nabla }_X{\partial }_r$ on the tangent space of a distance sphere. This is the negative of the shape operator with outward normal under the convention $S=-\nabla \nu$. A nonsingular matrix $Y$ of radial Jacobi fields satisfies $Y^{\prime \prime }+{\mathcal{R}}_{{\partial }_r}Y=0$ and $A=Y^{\prime }{Y}^{-1}$, yielding the displayed matrix Riccati equation. Taking its trace gives $h^{\prime }+\mathrm{tr}(A^2)+\mathrm{Ric}({\partial }_r,{\partial }_r)=0$, where $h=\mathrm{tr}A={\partial }_r\log \det Y$. This is the local differential identity behind the Bishop-Gromov inequality.

For $r$ smaller than the normal-coordinate radius, the geodesic circle on a Riemannian surface consists of points at distance $r$ from $p$. The exponential map restricted to the radius-$r$ circle in $T_{p}S$ gives its smooth embedded parameterization. Outside the injectivity range a distance sphere need not be smooth.

Fix $p$ on a Riemannian surface. For sufficiently small $r>0$, the geodesic circles of radius $r$ centered at $p$ and at every sufficiently nearby distinct $q$ meet transversally at exactly two points. In a strongly convex normal neighbourhood, write $q={\exp }_p(\epsilon v)$ and $x(\theta )={\exp }_p(re(\theta ))$. First variation givesThe quotient of the bracketed function by $\epsilon$ extends smoothly and has two simple zeros at zero. The implicit function theorem, uniformly over unit $v$, preserves these two zeros and excludes others. The nonzero angular derivatives prove transversality. The distinct-centre hypothesis is essential: coincident circles are not transverse.

The radial direction is orthogonal to the angular directions under the exponential map. On a surface this makes the first fundamental form in geodesic polar coordinateswith $\sqrt{G(r,\theta )}/r\to 1$ as $r\to 0$.

For $h(r,\theta )=\sqrt{G(r,\theta )}$, the angular Jacobi field equation on a surface becomesThe Riemannian area element is $h(r,\theta )drd\theta$.

If $h^{\prime \prime }+Kh=0$, $h(0)=0$, $h^{\prime }(0)=1$, $h>0$, and $K\le C$ for $C>0$, thenSubtract a smaller multiple of the sine solution and use the monotonicity of its Wronskian with $h$.

If $K\le 0$, then $h_{rr}=-Kh\ge 0$, so $h\ge r$. Every geodesic polar ball of radius $\epsilon$ therefore satisfies

If $K\le C$ with $C>0$, scalar comparison givesbefore $\pi /\sqrt{C}$. Consequently