I trying to translate some concepts to formalism of differential geometry.
Context
I started with a function $f:\mathbb R^{1+N}\rightarrow\mathbb R^m$ and I have defined the set $$ Q \equiv \{(t, x) \in \mathbb R^{1 + N} ~\vert~ f(t, x) = 0\}. $$
Under some conditions in $f$ we have that $Q$ is a n-manifold, with $n = 1 + N - m$.
- I have a time-dependent function $x:I\subset\mathbb R\rightarrow\mathbb R^N$ that induce a map $\gamma:I\rightarrow Q:t\mapsto(t, x(t))$ (ignore some complications in the set $I$).
- I have a function $U(t) = \frac{d}{dt} x(t)$ that induces the map $\vec v:I\rightarrow T(Q):t\mapsto\frac{d}{dt}\gamma(t)$. Since $T_{\gamma(t)}(Q)\neq\mathbb R^{1 + DN}$ then $\vec v(t) \neq (1, U(t))$.
- I have a linear map $M:\mathbb R^N\rightarrow\mathbb R^N$ that define the function $P(t) = M U(t)$.
My problem
I can't generalize $P$ to a vector tangent to $Q$.
What I'm trying to do (suggestion)
Provide a linear injection $\theta_a:T_a(Q)\rightarrow\mathbb R^{1+N}$ that make $T_a(Q)$ isomorfic to a linear subspace of $\mathbb R^{1+N}$ with dimension $n$, preferably such that $\theta_{\gamma(t)}(\vec v(t)) = (1, U(t))$. If I have such injection then I can define $\vec p(t) \equiv \theta_{\gamma(t)}^{-1}(m_0, MU(t))$ for some real constant $m_0$.
Another advantage of such injection is that I can induces in $Q$ a metric tensor $\eta$ computed in matrix notation as $$ \eta_a(X, Y) = \theta_a(X)~\operatorname{Id}_{\mathbb R}\otimes M~\theta_a(Y). $$
Note: There are no problem if for an arbitrary coordinates system $q^1, \cdots, q^{2n}$ defined in a neighborhood $U$ of $a\in T(Q)$ the injection $\theta_{\pi(a)}$ is dependent of this coordinate system. But is good if is coordinate independent.