Tangent space $T_p$ at a point $p$ on a manifold $M$.
Covariant tensor (共变张量) $a_\mu$ and contravariant tensor (反变张量) $a^\nu$, indicated with lowered and raised indices, are tensors which transform between coordinate systems $x_i'$ and $x_j$ according to $a_i' = a_i^j a_j$ and $a_i' = a_{ij} a_j$ respectively, where $a_i^j = \partial x_j / \partial x_i'$ and $a_{ij} = \partial x_i' / \partial x_j$.
Metric tensor (fundamental tensor, geometric structure) $g$ on a differentiable manifold $M$ is a twice covariant symmetric tensor field (张量场), where $g_p$ is its value at point $p$. The metric tensor completely determines the intrinsic geometry of the manifold.
a scalar product of contravariant vectors $x, y \in T_p$ $⟨x,y⟩$ bilinear function $g_p(x, y)$
Metric tensor $g_{\mu \nu}$ turns a contravariant tensor $a^\nu$ into a covariant tensor $a_\mu$: $g_{\mu \nu} a^\nu = a_\mu$. Metric tensor $g^{\mu \nu}$ turns a covariant tensor $a_\nu$ into a contravariant tensor $a^\mu$: $g^{\mu \nu} a_\nu = a^\mu$.
degenerate metric tensor
Riemannian metric tensor
pseudo-Riemannian metric tensor An example of non-Riemannian metric is the Minkowski metric of special relativity.
Exponential map $\exp{x}$.
The metric $d(x,y)$ of a complete Riemannian manifold is the length of the geodesic between two points.
Flat (Riemannian) manifold is a manifold with a Riemannian metric that has zero curvature, such that there is a coordinate system where the metric tensor is constant Kronecker delta $g_{\mu \nu} = \delta_{\mu \nu}$. Euclidean spaces are flat manifolds.
Symplectic geometry