Riemanian manifold

Riemannian manifold

• A smooth manifold $M$ with a Riemannaian metric $g$

Smooth manifold

• A type of manifold that is locally similar enough to a vector space to allow one to apply calculus.

Manifold

• A subspace that has similar features with higher dimension space.

Riemannian Metric

• A Riemannian metric $g$ on a smooth manifold $M$ is a smoothly chosen inner product $g_x : T_xM \times T_xM → R$ on each of the tangent spaces $T_xM$ of $M$. In other words, for each $x ∈ M, \;g = g_x$ satisfies
  (1). $g(u, v) = g(v,u),\;\forall u, v ∈ T_xM$ (symmetric)
  (2). $g(u, u) ≥ 0,\;\forall u ∈ T_xM$ (positive-definite)
  (3). $g(u, u) = 0\;\leftrightarrow \;u = 0$
  where $T_xM$ is the tangent space of $x$.

Satisfing inner product

• Letting $g(u,v) =⟨u, v⟩$,
  $⟨·, ·⟩: T_xM × T_xM → R$ satisfies inner product.

• Thus, there exsists a symmetric positive-definite matrix $M$, such that
  $⟨u, v⟩ = u^TMv\;\;$for all $\;u, v ∈ T_xM$.
  In this case, $⟨v, w⟩ = v^T g(x)w$