Tangent space $T_p M$ at a point $p$ on a manifold $M$. Exponential map $\exp{p}: T_p M \to M$.
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 tensor field (张量场) of rank-2, covariant, symmetric tensors. Its value $g_p$ at point $p \in M$ defines a scalar product $g_p(x, y)$ of contravariant vectors on the tangent space $T_p M$; in coordinate notation, $g_p(x, y) = g_{ij}(p) x^i y^j$. Tensor $g_p$ is regarded as an infinitesimal metric of the manifold $M$: the square of the differential arc length of curves in $M$ going from $p$ in direction $\mathrm{d}x$ is $\mathrm{d} s^2 = g_{ij}(p) \mathrm{d} x^i \mathrm{d} x^j$, which is called the metric form (or first fundamental form) of the metric tensor $g$. The metric tensor completely determines the intrinsic geometry of the manifold.
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 $a^\mu$: $g^{\mu \nu} a_\nu = a^\mu$.
A degenerate metric tensor is a metric tensor with $\det g_{ij} = 0$. An isotropic manifold is a manifold with a degenerate metric.
A Riemannian metric tensor is a metric tensor where $g_{ij}$ is positive definite. A pseudo-Riemannian metric tensor is a metric tensor where $g_{ij}$ is non-degenerate and indefinite. An example of non-Riemannian metric is the Minkowski metric of special relativity.
Riemannian space, Riemannian geometry, or Riemannian manifold, $(M, g)$ is an $n$-dimensional connected differentiable manifold $M$ with a Riemannian metric $g$. Riemannian geometry (see B. Riemann) is a multi-dimensional generalization of the intrinsic geometry of two-dimensional surfaces in three-dimensional Euclidean space (see C.F. Gauss). Riemannian geometry is essential for the general theory of relativity, and has wide application in mechanics and physics.
A Riemannian space is a metric space. A Riemannian space is called (geodesically) complete if it is complete as a metric space. The metric $d(x, y)$ between any two points of a complete Riemannian manifold is the length of a (not necessarily unique) geodesic between the points. Angle and volume are also defined on a Riemannian space by the Riemannian metric.
(Riemannian) normal coordinates.
Curvature. 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