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# 数学代写|微分几何代写DIFFERENTIAL GEOMETRY代写|Defnition and Gauß–Codazzi

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## 数学代写|微分几何代写DIFFERENTIAL GEOMETRY代写|Defnition and Gauß–Codazzi

Let $M \subset \mathbb{R}^n$ be a smooth manifold and $\gamma: \mathbb{R}^2 \rightarrow M$ be a smooth map. Denote by $(s, t)$ the coordinates on $\mathbb{R}^2$. Let $Z \in \operatorname{Vect}(\gamma)$ be a smooth vector field along $\gamma$, i.e. $Z: \mathbb{R}^2 \rightarrow \mathbb{R}^n$ is a smooth map such that $Z(s, t) \in T_{\gamma(s, t)} M$ for all $s$ and $t$. The covariant partial derivatives of $Z$ with respect to the variables $s$ and $t$ are defined by
$$\nabla_s Z:=\Pi(\gamma) \frac{\partial Z}{\partial s}, \quad \nabla_t Z:=\Pi(\gamma) \frac{\partial Z}{\partial t} .$$
In particular $\partial_s \gamma=\partial \gamma / \partial s$ and $\partial_t \gamma=\partial \gamma / \partial t$ are vector fields along $\gamma$ and we have $\nabla_s \partial_t \gamma-\nabla_t \partial_s \gamma=0$ as both terms on the left are equal to $\Pi(\gamma) \partial_s \partial_t \gamma$. Thus ordinary partial differentiation and covariant partial differentiation commute. The analogous formula (which results on replacing $\partial$ by $\nabla$ and $\gamma$ by $Z$ ) is in general false. Instead we have the following.

Definition 5.2.1 The Riemann curvature tensor assigns to each $p \in M$ the bilinear map $R_p: T_p M \times T_p M \rightarrow \mathcal{L}\left(T_p M, T_p M\right)$ characterized by the equation
$$R_p(u, v) w=\left(\nabla_s \nabla_t Z-\nabla_t \nabla_s Z\right)(0,0)$$
for $u, v, w \in T_p M$ where $\gamma: \mathbb{R}^2 \rightarrow M$ is a smooth map and $Z \in \operatorname{Vect}(\gamma)$ is a smooth vector field along $\gamma$ such that
$$\gamma(0,0)=p, \quad \partial_s \gamma(0,0)=u, \quad \partial_t \gamma(0,0)=v, \quad Z(0,0)=w$$

## 数学代写|微分几何代写DIFFERENTIAL GEOMETRY代写|Covariant Derivative of a Global Vector Field

So far we have only defined the covariant derivatives of vector fields along curves. The same method can be applied to global vector fields. This leads to the following definition.

Definition 5.2.3 (Covariant derivative) Let $M \subset \mathbb{R}^n$ be an $m$-dimensional submanifold and $X$ be a vector field on $M$. Fix a point $p \in M$ and a tangent vector $v \in T_p M$. The covariant derivative of $X$ at $p$ in the direction $v$ is the tangent vector
$$\nabla_v X(p):=\Pi(p) d X(p) v \in T_p M,$$
where $\Pi(p) \in \mathbb{R}^{n \times n}$ denotes the orthogonal projection onto $T_p M$.
Remark 5.2.4 Let $\gamma: I \rightarrow M$ be a smooth curve on an interval $I \subset \mathbb{R}$ and let $X \in \operatorname{Vect}(M)$ be a smooth vector field on $M$. Then $X \circ \gamma$ is a smooth vector field along $\gamma$ and the covariant derivative of $X \circ \gamma$ is related to the covariant derivative of $X$ by the formula
$$\nabla(X \circ \gamma)(t)=\nabla_{\dot{\gamma}(t)} X(\gamma(t))$$
Remark 5.2.5 (Gauß-Weingarten formula) Differentiating the equation $X=$ $\Pi X$ (understood as a function from $M$ to $\mathbb{R}^n$ ) and using the notation $\partial_v X(p):=$ $d X(p) v$ for the derivative of $X$ at $p$ in the direction $v$ we obtain the GaußWeingarten formula for global vector fields:
$$\partial_v X(p)=\nabla_v X(p)+h_p(v) X(p)$$

## 数学代写|微分几何代写DIFFERENTIAL GEOMETRY代写|Defnition and Gauß–Codazzi

$$\nabla_s Z:=\Pi(\gamma) \frac{\partial Z}{\partial s}, \quad \nabla_t Z:=\Pi(\gamma) \frac{\partial Z}{\partial t}$$

$$R_p(u, v) w=\left(\nabla_s \nabla_t Z-\nabla_t \nabla_s Z\right)(0,0)$$

$$\gamma(0,0)=p, \quad \partial_s \gamma(0,0)=u, \quad \partial_t \gamma(0,0)=v, \quad Z(0,0)=w$$

## 数学代写|微分几何代写DIFFERENTIAL GEOMETRY代写|Covariant Derivative of a Global Vector Field

$$\nabla_v X(p):=\Pi(p) d X(p) v \in T_p M$$

$$\nabla(X \circ \gamma)(t)=\nabla_{\dot{\gamma}(t)} X(\gamma(t))$$

$$\partial_v X(p)=\nabla_v X(p)+h_p(v) X(p)$$

## MATLAB代写

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