Physical Object to Pseudo-Riemannian Manifold It is well known that Lorentzian manifolds are studied in general relativity. So this raises my curiosity about:

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*How about classical mechanics? Does it correspond to the manifold $\mathbb{R}\times M$, and uses the $(n,0)$-metric instead of $(n-1,1)$?


*More generally, what is the physical object to the pseudo-Riemannian geometry?
Any advice is helpful. Thank you.
 A: The reason why nonpositive metrics are introduced in relativity comes from the simple following observation: if a particle with positive mass moves between two points in Euclidean space $\Bbb R^3$ with displacement vector $(\Delta x, \Delta y, \Delta z)$, over some time interval $\Delta t > 0$, then the fact that the particle could not have moved with a speed greater than of the speed of light in vacuum $c$ reads $$\left(\frac{\Delta x}{\Delta t}\right)^2 + \left(\frac{\Delta y}{\Delta t}\right)^2 + \left(\frac{\Delta z}{\Delta t}\right)^2 < c^2, \tag{1}$$which may be reorganized as $$(\Delta x)^2+(\Delta y)^2 + (\Delta z)^2 - c^2(\Delta t)^2 < 0,\tag{2}$$whose left hand side is manifestly quadratic in the variables $\Delta x$, $\Delta y$, $\Delta z$ and $\Delta t$. Polarizing this expression gives rise to the Minkowski scalar product $$\langle v,w\rangle_L = v_1w_1+v_2w_2+v_3w_3 - v_4w_4.\tag{4}$$
Pseudo-Riemannian manifolds are simply a geometric generalization of Riemannian and Lorentzian manifolds, allowing multiple orthogonal timelike directions. However, some spacetimes (such as the anti-de Sitter space) appear as submanifolds of pseudo-Riemannian manifolds whose metric has index higher than $1$ (with the induced metric being Lorentzian, in the same fashion that submanifolds of Lorentzian manifolds may happen to be Riemannian).
As for classical mechanics, the point is that one usually considers a "configuration manifold" $Q$, without initially having any metric equipped on it, but it gives rise to a "state space" of position and momenta: its cotangent bundle $T^*Q$. The cotangent bundle has a very natural structure: writing $\pi\colon T^*Q\to Q$ for the projection taking $(x,p)\mapsto x$ (here $p\in T_x^*Q$), one may define a $1$-form $\lambda$ on $T^*Q$ by $$\lambda_{(x,p)}(Z) = p({\rm d}\pi_{(x,p)}(Z)),\qquad Z\in T_{(x,p)}(T^*Q).\tag{5}$$Relative to cotangent coordinates $(q^1,\ldots, q^n,p_1,\ldots, p_n)$ induced on $T^*Q$ by coordinates $(q^1,\ldots, q^n)$, one has that $$\lambda = p_1\,{\rm d}q^1+\cdots + p_n\,{\rm d}q^n,\tag{6}$$and hence $$\omega \doteq -{\rm d}\lambda = {\rm d}q^1\wedge {\rm d}p_1+\cdots + {\rm d}q^n\wedge {\rm d}p_n\tag{7}$$is a closed and non-degenerate $2$-form on $T^*Q$. A Hamiltonian function $H\colon T^*Q \to \Bbb R$ (i.e., a function of position and momentum which controls the time evolution of your mechanical system) gives rise via $\omega$ to a vector field $X_H$ on $T^*Q$, characterized by $\omega(X_H,\cdot) = {\rm d}H$. In other words, the natural tensor for studying classical mechanics is not symmetric, but instead skew-symmetric, and Hamiltonian mechanics is encoded in the flow of a vector field. The abstract generalization of this setup is the one of a symplectic manifold: a pair $(M,\omega)$ where $M$ is a smooth manifold and $\omega$ is a closed and non-degenerate $2$-form. There is a very extensive literature on this and more.
