There is a symbol in the book Geometric Relativity by Dan A. Lee that confuses me.
We will fix a specific submanifold $\Sigma^m\subseteq M^n$, and take $\mathcal{M}$ to be $\mathrm{Diff}_0(M)$, the space of all diffeomorphisms of $M$ in the same component as the identity. By pushing forward $\Sigma$ via diffeomorphism, this space will parameterize all of the submanifolds of $M$ that are isotopic to $\Sigma$ (indeed, this is the definition of isotopic). Of course, this parameterization introduces a huge amount of redundancy, but these redundancies will not cause problems for us. Even without the formalism of infinite-dimensional manifolds, we can still think intuitively about what the "tangent space" of $\mathrm{Diff}_0(M)$ at the identity should be: the space of smooth vector fields $C^\infty(TM)$. Explicitly, if one considers a smooth path $\Phi:(−\epsilon,\epsilon)\to\mathrm{Diff}_0(M)$ such that $\Phi_0$ is the identity, then we can define a vector field $X(p) = \frac{\partial}{\partial t}|_{t=0} \Phi_t(p)$ for all $p\in M$. More generally, we can define $X_t(\Phi_t(p)) = \frac{\partial}{\partial t}\Phi_t(p)$ so that $X_0 = X$. We abbreviate this by writing $X_t = \frac{\partial}{\partial t}\Phi_t$. We often refer to the family $\Phi_t$ as a one-parameter family of deformations and $X_t$ as its first-order deformation vector field.
Fix a compact submanifold $\Sigma$ of a Riemannian manifold $(M,g)$ with induced metric $h$ and induced volume measure $d\mu_\Sigma = d\mu_h$. Define $\Sigma_t =\Phi_t(\Sigma)$, where $\Phi_t$ is as described above. Our first goal is to compute the linearization of the volume functional at $\Sigma$ in the direction of $X$, also called the first variation of volume with respect to the first-order deformation $X$, which is just the quantity $\frac{d}{dt}|_{t=0} \mu(\Sigma_t)$.
The symbol $M$ should denote a Riemannian manifold. I was wondering the exact meaning of $\mathrm{Diff}_0(M)$. What are those diffeomorphisms of $M$ in the same component as the identity? What does the component mean? And what does it mean to be isotopic? Is it related to algebraic geometry? Is there an introductory book for people without knowledge of algebra and algebraic geometry?
Thank you.