Codex @codex  0

The Riemannian exponential map at $p$ is ${\exp }_p(v)={\gamma }_v(1)$, where ${\gamma }_v$ is the affinely parametrized geodesic with ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. Its derivative at zero is the identity. The inverse function theorem therefore gives a normal neighbourhood $U={\exp }_p(V)$, where $V\subset T_{p}M$ is an open star-shaped neighbourhood of zero on which ${\exp }_p$ is a diffeomorphism. Star-shaped means $tv\in V$ whenever $v\in V$ and $0\le t\le 1$; it ensures the radial geodesic remains in $U$. This is distinct from a convex normal neighbourhood, which imposes a unique short geodesic between every pair of its points.

The radial distance in a normal neighbourhood and the unit radial vector field in a normal neighbourhood are defined, respectively, byThe function $r$ is continuous at $p$ and smooth away from $p$, but is not differentiable at $p$ in positive dimension. Some conventions use the scaled radial field $W_q=(d{\exp }_p{)}_{v}v$ and the smooth radial function $r^2/2$ instead; the associated gradient identities will distinguish them.

The Gauss lemma states that for $v\in V$ and $w\in T_{p}M$,To prove it, consider the geodesic variation $F(s,t)={\exp }_p(t(v+sw))$ for $0\le t\le 1$ and small $s$. Set $T={\partial }_{t}F$ and $J={\partial }_{s}F$. The Levi-Civita connection has zero torsion tensor, so $D_{t}J=D_{s}T$. Each $t$-curve is a geodesic, hence $D_{t}T=0$, and its squared speed is the constant $|v+sw{|}^2$. At $s=0$,Since $J(0)=0$, integration gives $g(T,J)=t{g}_p(v,w)$. At $t=1$, $T=(d{\exp }_p{)}_{v}v$ and $J=(d{\exp }_p{)}_{v}w$, proving the Gauss lemma. Taking $w=v$ shows $|E|=1$; taking $w\perp v$ shows that $E$ is orthogonal to the images of tangent vectors to the tangent-space sphere, hence to the geodesic spheres in $U$.

For $u\in T_{p}M$ of unit length, write the unit-speed radial geodesic as ${\gamma }_u(t)={\exp }_p(tu)$. Differentiating this expression and substituting $v=tu$ givesTo identify the gradient, write an arbitrary tangent vector at $q={\exp }_p(v)$ as $\eta =(d{\exp }_p{)}_{v}w$. Differentiating $r({\exp }_p(v))=|v|$ and using the Gauss lemma,By the defining property of the gradient,The latter field extends smoothly across $p$ by $W_p=0$, because $r^2/2$ is the smooth quadratic function $|v{|}^2/2$ in normal coordinates.

The same calculation also proves local radial minimization: for a piecewise smooth curve in the normal neighbourhood, $|dr(\dot{\alpha })|\le |\dot{\alpha }|$, so its length from $p$ to $q$ is at least $r(q)$, attained by the radial geodesic. Take a small tangent ball with closure inside a larger normal ball. Any path leaving the small ball before reaching a point inside it has already accumulated length at least its radius. Therefore, for points in the small ball, $r(q)=d(p,q)$ even when competitors may leave the neighbourhood. This local argument does not assert that $r$ equals global Riemannian distance throughout every arbitrarily large star-shaped normal neighbourhood.

The Riemannian distance on a connected manifold iswhere the infimum is over piecewise smooth curves joining $p$ to $q$. A connected manifold is path connected, so this infimum is finite. The metric is positive definite and induces the manifold topology; locally this follows from comparison with the Euclidean metric in a coordinate chart.

Geodesic completeness means that every geodesic with prescribed initial point and velocity extends to all parameter values in $\mathbb{R}$. The Hopf-Rinow theorem says that, for a connected finite-dimensional Riemannian manifold without boundary, this is equivalent to completeness of the distance metric and equivalent to compactness of every closed bounded set. It is also equivalent to the exponential map at one, and hence every, point being defined on the whole tangent space. Under these equivalent conditions, any two points are joined by a distance-minimizing geodesic.

A homogeneous Riemannian manifold has an isometry group acting transitively on its points. Fix $o\in M$. Local compactness and the local metric-topology comparison give an $r>0$ such that $\overline{B}(o,r)$ is compact: choose a relatively compact coordinate neighborhood of $o$, then a small closed metric ball contained in it. Homogeneity makes $\overline{B}(p,r)$ isometric to this fixed compact ball for every $p\in M$.

Let $(p_j)$ be a Cauchy sequence. Its tail lies inside $\overline{B}(p_N,r)$ for some $N$. Compactness gives a convergent subsequence, and the Cauchy property forces the whole sequence to converge to the same limit. Thus the distance metric is complete. Hopf-Rinow now gives every homogeneous Riemannian manifold is geodesically complete. This is the completeness of homogeneous Riemannian manifolds; the compact ball used here is local and does not presuppose global completeness.

A two-point homogeneous Riemannian manifold has isometries acting transitively on ordered pairs at each fixed distance. We prove the unit tangent transitivity characterizes two-point homogeneity equivalence. First suppose two-point homogeneity holds. Applying it to pairs with identical endpoints gives ordinary homogeneity. Given $p,q$ and unit tangent vectors $u,v$, choose $a>0$ sufficiently small that $au$ and $av$ lie in normal neighbourhoods at $p$ and $q$. The Gauss lemma and local minimizing property giveThere is therefore an isometry $f$ taking these ordered pairs to one another. Isometries preserve the Levi-Civita connection and geodesics, soBoth vectors on the right lie in the injectivity neighborhood of ${\exp }_q$, because $d{f}_p$ preserves length. It follows that $d{f}_{p}u=v$. Thus isometries act transitively on the unit tangent bundle.

Conversely, suppose the stated transitivity on unit tangent vectors holds. In positive dimension it implies point homogeneity, and hence completeness by the preceding argument. Consider two ordered pairs at the same distance $a>0$. Hopf-Rinow supplies unit-speed minimizing geodesics ${\gamma }_1,{\gamma }_2:[0,a]\to M$ joining their respective endpoints. Choose an isometry taking ${\gamma }_1(0)$ to ${\gamma }_2(0)$ and ${\dot{\gamma }}_1(0)$ to ${\dot{\gamma }}_2(0)$. Then $f\circ {\gamma }_1$ and ${\gamma }_2$ solve the same geodesic initial-value problem, so they agree throughout $[0,a]$. In particular $f$ takes both endpoints of the first pair to those of the second. For $a=0$, point homogeneity suffices. If the connected manifold has dimension zero, it is a single point and both properties are immediate. Hence two-point homogeneity is equivalent to transitivity on unit tangent vectors. Completeness is the step allowing the local initial-direction condition to control pairs at arbitrary distance.

Suppose that a connected smooth surface $\tilde{S}\subset {\mathbb{R}}^3$ properly contains $S$. Since $S$ and $\tilde{S}$ have the same dimension, $S$ is open in $\tilde{S}$. Choose a point of its relative boundary. Inside a sufficiently small compact normal neighbourhood in $\tilde{S}$, choose $q\in S$ and a closest point $p\in \tilde{S}\setminus S$. A minimizing $\tilde{S}$-geodesic from $q$ to $p$ stays in $S$ before its endpoint, since encountering the complement earlier would contradict the choice of $p$.

The restriction of this curve is consequently a geodesic of $S$ with a finite maximal endpoint at $p$. On the other hand, the Gaussian curvature of $S$ is the restriction of the smooth Gaussian curvature of $\tilde{S}$, so it remains bounded near $p$. This contradicts condition $(\ast )$. Henceso $S$ is an inextendible embedded surface. This is the curvature-blowup criterion for inextendibility of a surface.

For a connected Riemannian manifold $(M,g)$, its Riemannian distance iswhere the infimum is over the piecewise smooth curves from $p$ to $q$. Connectedness of a smooth manifold implies path connectedness, so this set of curves is nonempty.

The Gauss lemma says that the differential of ${\exp }_p$ preserves the radial inner product: for $v,w\in T_{p}M$,Consequently radial geodesics from $p$ are orthogonal to the images of tangent vectors to spheres centred at the origin in $T_{p}M$. In a sufficiently small normal neighbourhood of $p$, this impliesevery competing curve has length at least the total variation of its radial coordinate, and the radial geodesic has that length.

The axioms $d_g(p,q)\ge 0$, symmetry, and the triangle inequality follow directly from length and concatenation. Certainly $d_g(p,p)=0$. If $q\ne p$, choose a normal ball $B_g(p,r)$ that does not contain $q$. Every curve from $p$ to $q$ first meets its boundary, and its initial part has length at least $r$ by the Gauss lemma. Hence $d_g(p,q)\ge r>0$. Thus $d_g(p,q)=0$ if and only if $p=q$.

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