diff --git a/content/Math401/Math401_P1_2.md b/content/Math401/Math401_P1_2.md
index 218899e..8f17cf0 100644
--- a/content/Math401/Math401_P1_2.md
+++ b/content/Math401/Math401_P1_2.md
@@ -66,7 +66,9 @@ The Haar measure is the unique probability measure that is invariant under the a
 
 _The existence and uniqueness of the Haar measure is a theorem in compact lie group theory. For this research topic, we will not prove it._
 
-### Random sampling on the $CP^n$
+### Random sampling on the $\mathbb{C}P^n$
+
+Note that the space of pure state in bipartite system
 
 ## Statement
 
diff --git a/content/Math401/Math401_P1_3.md b/content/Math401/Math401_P1_3.md
index a4b53e2..d290134 100644
--- a/content/Math401/Math401_P1_3.md
+++ b/content/Math401/Math401_P1_3.md
@@ -1,5 +1,20 @@
 # Math 401 Paper 1, Side note 3: Levy's concentration theorem
 
+Our goal is to prove the generalized version of Levy's concentration theorem used in Hayden's work for $\eta$-Lipschitz functions.
+
+Let $f:S^{n-1}\to \mathbb{R}$ be a $\eta$-Lipschitz function. Let $M_f$ denote the median of $f$ and $\langle f\rangle$ denote the mean of $f$. (Note this can be generalized to many other manifolds.)
+
+Select a random point $x\in S^{n-1}$ with $n>2$ according to the uniform measure (Haar measure). Then the probability of observing a value of $f$ much different from the reference value is exponentially small.
+
+$$
+\operatorname{Pr}[|f(x)-M_f|>\epsilon]\leq \exp(-\frac{n\epsilon^2}{2\eta^2})
+$$
+$$
+\operatorname{Pr}[|f(x)-\langle f\rangle|>\epsilon]\leq 2\exp(-\frac{(n-1)\epsilon^2}{2\eta^2})
+$$
+
+> This version of Levy's concentration theorem can be found in [Geometry of Quantum states](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C) 15.84 and 15.85.
+
 ## Basic definitions
 
 ### Lipschitz function
@@ -200,6 +215,8 @@ We use the Maxwell-Boltzmann distribution law and Levy's isoperimetric inequalit
 
 The goal is the same as the Gromov's version, first we bound the probability of the sub-level set of $f$ by the $\kappa_n(\epsilon)$ function by Levy's isoperimetric inequality. Then we claim that the $\kappa_n(\epsilon)$ function is bounded by the Gaussian distribution.
 
+Note, this section is not rigorous enough in sense of mathematics and the author should add sections about Levy family and observable diameter to make the proof more rigorous and understandable.
+
 <details>
 <summary>Proof</summary>
 
@@ -219,8 +236,6 @@ $$
 \operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))\leq \operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_+))+\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_-))
 $$
 
-
-
 We define $\kappa_n(\epsilon)$ as the following:
 
 $$
@@ -240,8 +255,41 @@ $$
 
 Not very edible for undergraduates.
 
+## Crash course on Riemannian manifolds
+
+> This part might be extended to a separate note, let's check how far we can go from this part.
+>
+> References:   
+>
+> - [Riemannian Geometry by John M. Lee](https://www.amazon.com/Introduction-Riemannian-Manifolds-Graduate-Mathematics/dp/3319917544?dib=eyJ2IjoiMSJ9.88u0uIXulwPpi3IjFn9EdOviJvyuse9V5K5wZxQEd6Rto5sCIowzEJSstE0JtQDW.QeajvjQEbsDmnEMfPzaKrfVR9F5BtWE8wFscYjCAR24&dib_tag=se&keywords=riemannian+manifold+by+john+m+lee&qid=1753238983&sr=8-1)
+
 ### Riemannian manifolds
 
+A Riemannian manifold is a smooth manifold equipped with a **Riemannian metric**, which is a smooth assignment of an inner product to each tangent space $T_pM$ of the manifold.
+
+An example of Riemannian manifold is the sphere $\mathbb{C}P^n$.
+
+### Riemannian metric
+
+A Riemannian metric is a smooth assignment of an inner product to each tangent space $T_pM$ of the manifold.
+
+An example of Riemannian metric is the Euclidean metric on $\mathbb{R}^n$.
+
+### Notion of Connection
+
+A connection is a way to define the directional derivative of a vector field along a curve on a Riemannian manifold.
+
+For every $p\in M$, where $M$ denote the manifold, suppose $M=\mathbb{R}^n$, then let $X=(f_1,\cdots,f_n)$ be a vector field on $M$. The directional derivative of $X$ along the point $p$ is defined as
+
+$$
+D_VX=\lim_{h\to 0}\frac{X(p+h)-X(p)}{h}
+$$
+
+### Nabla notation and Levi-Civita connection
+
+
+### Ricci curvature
+
 
 
 ## References