A degenerate pullback metric under a constant map

Given: $F:M\to \mathbb{R}$ is constant, $M$ is a smooth manifold of dimension $n>0$, and the target metric is $d{y}^2$.

Pullback of a riemannian metric as a tensor: For smooth $F:M\to N$ and a Riemannian metric $h$ on $N$, its pullback tensor is $(F^{\ast }h{)}_p(v,w)=h_{F(p)}(d{F}_{p}v,d{F}_{p}w)$. This is def-pullback-of-a-covariant-tensor-field for the tensor in def-riemannian-metric-and-riemannian-manifold. It is always symmetric and positive semidefinite; the name does not assert positive definiteness. Smoothness and the precise immersion criterion are established next.

Pullback of a riemannian metric is riemannian exactly for immersions: $F^{\ast }h$ is Riemannian if and only if $F$ is an immersion. In general it is positive semidefinite, with radical $\ker d{F}_p$ at $p$.

Smooth manifolds and their smooth charts: A smooth $n$-manifold is a pair $(M,\mathcal{S})$ in which $M$ is a topological $n$-manifold (def-topological-manifold-without-boundary) and $\mathcal{S}$ is a smooth structure on $M$: a maximal smooth atlas (thm-each-smooth-atlas-is-contained-in-a-unique-maximal-smooth-atlas). Because thm-each-smooth-atlas-is-contained-in-a-unique-maximal-smooth-atlas sends every smooth atlas to the unique maximal atlas containing it, a smooth manifold is equivalently specified by a topological manifold $M$ together with any one smooth atlas $\mathcal{A}$, the structure being the generated $[\mathcal{A}]$. A chart $(U,\phi )\in \mathcal{S}$ is called a smooth chart (or a chart of the smooth structure); its domain is a coordinate domain and its coordinate functions are smooth coordinates on $U$. When the structure is clear from context, the manifold itself is written $M$ in place of $(M,\mathcal{S})$.

Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces: Let $n\in \mathbb{N}$. For $n\ge 1$, put ${\mathbb{R}}^n:={\prod }_{k<n}\mathbb{R}$ with its usual topology; for $n=0$ put ${\mathbb{R}}^0:=\{0\}$, the one-point space. A topological $n$-manifold without boundary (or briefly an $n$-manifold) is a topological space $M$ satisfying: 1. $M$ is Hausdorff (def-hausdorff-space); 2. $M$ is second countable (def-second-countable-space); 3. $M$ is locally Euclidean of dimension $n$: every $p\in M$ has an open neighbourhood $U\subseteq M$ homeomorphic to an open subset of ${\mathbb{R}}^n$ (def-homeomorphism-and-open-maps). The empty space satisfies all three conditions vacuously, so $M=\mathrm{\varnothing }$ is an $n$-manifold for every $n$; this degenerate instance is kept, and statements about nonempty manifolds name the hypothesis. In dimension zero, condition 3 forces the one-point neighbourhoods of points to be open singletons, so a $0$-manifold is exactly a discrete second-countable space with at most countably many points.