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If a smooth map has constant rank $r$ near a point, there are local coordinates in which it is

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 finite-dimensional commutative real algebra with a Euclidean multiplicative norm has dimension at most two. Automatic real bilinearity from a multiplicative norm supplies bilinearity if it was not part of the initial description. On its unit sphere, squaring has derivative $v\mapsto 2xv$, injective because there are no zero divisors. The only equal squares are antipodal pairs, so the map induces an injective immersion ${\mathbb{RP}}^{n-1}\to S^{n-1}$. Compactness and the inverse function theorem make it a diffeomorphism when $n\ge 2$. For $n\ge 3$, the fundamental groups $\mathbb{Z}/2$ and zero contradict this. The real and complex number algebras attain dimensions one and two.

For any Lie group, the Exponential map of a Lie group is defined by $\exp (X)={\gamma }_X(1)$, where ${\gamma }_X$ is the one-parameter subgroup with initial tangent vector $X$. For a Matrix Lie group, it is the matrix exponential $\exp X={\sum }_{n\ge 0}{X}^n/n!$. Its differential at zero is the identity, so the inverse function theorem supplies a local exponential chart. A small transformation can therefore be encoded by a vector in the Lie algebra. See MIT 18.745, Lecture 7, Definition 7.2 and Theorem 7.5.

Local invertibility makes no promise about a global inverse. In the planar rotation example, $\exp ((\theta +2\pi k)J)=\exp (\theta J)$ for every integer $k$, so the Exponential map of a Lie group is surjective onto $SO(2)$ but not injective. Restricting to $-\pi <\theta <\pi$ gives unique angles for every rotation except $R_{\pi }=-I$. The omitted half-turn exposes the cut needed for this single angle chart.

Surjectivity can also fail, even in a connected Lie group. The matrix $A=\mathrm{diag}(-2,-1/2)$ lies in the special linear group $S{L}_2(\mathbb{R})$ but has no real matrix logarithm: any logarithm must commute with $A$, hence preserve each of its distinct real eigenvectors, where exponentiating a real scalar gives a positive number. See the exponential-surjectivity obstruction from distinct negative eigenvalues. These two examples distinguish two different global obstacles: repeated parameters and unreachable elements. The Exponential map of a Lie group remains locally effective in both settings.

If $F(x_0,y_0)=0$ and the partial derivative with respect to $y$ is invertible at $(x_0,y_0)$, then the nearby zero set is the graph $y=g(x)$ of a continuously differentiable function.

Let $f:M\to N$ be a continuous map between oriented $d$-dimensional manifolds, with $d\ge 1$. Suppose $x$ is an isolated point of the fiber over $y=f(x)$. Choose an open neighborhood $U$ with $U\cap f^{-1}(y)=\{x\}$. The map of pairsinduces a map between rank-one degree-$d$ local homology groups. Relative to the local orientation of a manifold, this map is multiplication by an integer, the local degree. Excision makes it independent of the neighborhood. A smooth map between manifolds with invertible derivative at $x$ has local degree $\mathrm{sign}\det D{f}_x$, by the inverse function theorem and the chosen orientations.

A local diffeomorphism restricts near every point to a diffeomorphism onto an open subset. In particular, every local diffeomorphism is an open map.

Since the differential of the Exponential map of a Lie group at zero is the identity, the inverse function theorem makes it a diffeomorphism on a sufficiently small neighborhood of zero. Its local inverse is a logarithm near the group identity. It need not extend to a global logarithm, even for the real general linear group.

The velocity of a geodesic can be extended to one smooth ambient vector field on a sufficiently short neighbourhood of any parameter value. If its initial velocity is zero, uniqueness of the geodesic equation makes it locally constant, and the zero field works. Otherwise one coordinate has nonzero velocity; the inverse function theorem supplies coordinates straightening the small embedded arc to $(t,0,\dots ,0)$. The coordinate field $\partial /\partial t$ is the required extension. Shrinking the interval is essential: arbitrary fields along self-intersecting curves need not have one ambient extension.

A near-identity transformation is a local change of variables whose derivative at the reference point is the identity. In a normal form of a dynamical system, substituting $x=y+h(y)$ and solving $(I+Dh(y))\dot{y}=f(y+h(y))$ determines which nonlinear coefficients can be removed. The transformation is locally invertible by the inverse function theorem. Terms whose removal equations are singular are resonant and must be retained; a computed zero resonant coefficient cannot be assigned an arbitrary nonzero value by normalization.

For $a{u}_{x_1}+b{u}_{x_2}=c$ with data along $(\xi (s),\eta (s))$, the initial curve is noncharacteristic when its tangent is linearly independent of the projected characteristic vector $(a,b)$ evaluated at the initial value of $u$. The determinant is $a{\eta }^{\prime }-b{\xi }^{\prime }$, up to sign. For $C^1$ coefficients and data this permits the inverse function theorem to invert the local characteristic parametrization and gives a unique local classical solution.

An open map sends every open set to an open set.