---- $\nabla^{N}_{X}Y= \nabla_{X}^{M}Y+\textcolor{Thistle}{\mathrm{II}(X, Y)}$ > [!definition] Definition. ([[second fundamental form]]) > Consider a [[Riemannian manifold|Riemannian]] [[embedded submanifold]] $(M, \iota^{*} g) \subset_{\iota} (N,g)$. The [[tangent bundle]] $TN$ [[direct sum|splits]] [[pullback of a vector bundle|along]] $M$ as the [[orthogonal direct sum of modules|orthogonal direct sum]] $TN |_{M}=TM \oplus (TM)^{\perp},$ > where $(TM)^{\perp}=\nu_{M \subset N}$ denotes the [[normal bundle]] of $M$ in $N$. If $X \in \Gamma(TN |_{M})$ is a [[vector field]] on $N$ along $M$, we write $X=\underbrace{ X^{\perp} }_{ \in \Gamma(\nu_{M \subset N}) }+\underbrace{ X^{T} }_{ \in \Gamma(TM) }$ > [[orthogonal projection|for its decomposition]] into components [[normal bundle|normal]] and tangential to $M$. [[partial covariant derivative|The]] [[Levi-Civita connection]] $\nabla^{N}: \Gamma(TN) \times \Gamma(TN) \to \Gamma(TN)$ on $N$ decomposes thus according to the **Gauss formula** $\nabla^{N}_{X} Y=\textcolor{Thistle}{{ (\nabla^{N}_{X}Y )^{\perp} }_{ }} +\textcolor{Skyblue}{\underbrace{ (\nabla_{X}^{N}Y)^{T} }_{ = \nabla _{X}^{M} Y }} ,$ > where the identification of $\textcolor{Skyblue}{(\nabla_{X}^{N}Y)^{T}}$ with $\textcolor{Skyblue}{\nabla_{X}^{M}Y}$ implicitly extends $X,Y \in \Gamma(TM)$ arbitrarily to vector fields on the whole of $N$.[^1] The $\nu_{M \subset N}$-[[differential form with values in a vector bundle|valued]] [[symmetric power|symmetric]] $(0,2)$-[[tensor product of vector bundles|tensor field]] $\textcolor{Thistle}{\mathrm{I I}}\in \operatorname{Sym}^{2}( T^{*} M ) \otimes \nu_{M \subset N}$,[^2] $\begin{align} > \textcolor{Thistle}{\mathrm{I I}: \Gamma(TM) \times \Gamma(TM) } &\textcolor{Thistle}{ \to \Gamma(\nu_{M \subset N})} \\ > \textcolor{Thistle}{(X, Y) } & \textcolor{Thistle}{\mapsto } \textcolor{Thistle}{(\nabla^{N}_{X}Y)^{\perp}} = \nabla_{X}^{N} Y - \textcolor{Skyblue}{\nabla_{X}^{M}Y} > \end{align}$ > is called the **second fundamental form of $M \subset N$**. It measures the discrepancy between the ambient connection $\nabla^{N}$ restricted to the [[embedded submanifold|submanifold]] $M$ and the [[embedded submanifold|submanifold]] $Ms [[Riemannian manifold|metric]] [[connection on a manifold|connection]] $\nabla^{M}$. > > > Contracting $\textcolor{Thistle}{\mathrm{I I}}$ with a [[normal bundle|normal vector field]] $\xi \in \Gamma(\nu)$ via the [[metric tensor|metric]] $g |_{\nu}$ determines a (scalar) [[symmetric multilinear map|symmetric]] $(0,2)$-[[tensor product of vector bundles|tensor field]] $h_{\xi}$ on $M$ as $h_{\xi}(X, Y)=g\big(\textcolor{Thistle}{\mathrm{I I}(X, Y)}, \xi \big).$We call $h_{\xi}$ the **second fundamental form in the direction of $\xi$**. > > Since $g$ is [[nondegenerate bilinear form|nondegenerate]], to $\xi$ there corresponds a unique [[category|endomorphism]] $S_{\xi}:TM \to TM$ such that[^7] $g(S_{\xi}X , Y)=h_{\xi}(X, Y)=g\big( \textcolor{Thistle}{\mathrm{ I I}(X,Y)}, \xi \big).$ > $S_{\xi}$ is called the **shape operator** or **shape endomorphism** associated to the [[normal bundle|normal field]] $\xi$. It follows from symmetry of $\textcolor{Thistle}{\mathrm{ I I}}$ that $S_{\xi}$ is [[self-adjoint]]. One derives the **Weingarten formula**[^10] $S_{\xi}(X)=-(\nabla_{X}^{N}\xi)^{T} . $ SO: "how does the second fundamental form interact with the metric in the direction of the normal vecotr $\xiquot; [^10]: This follows from the computation $\begin{align} h_{\xi}(X, Y)=g\big( S_{\xi}X, Y \big) &= g\big(\textcolor{Thistle}{ \mathrm{II}(X, Y)}, \xi \big) \\ &= g\big( \textcolor{Thistle}{(\nabla_{X}^{N}Y)^{\perp}}, \xi \big) \\ &= g(\nabla_{X}^{N}Y, \xi) \, \, (\text{using that }\xi \text{ is normal}) \\ &= \underbrace{X{ g(Y, \xi) } }_{ =0 \text{ since }Y=Y^{\top}, \xi=\xi^{\perp} } - g(\nabla_{X}^{N}\xi, Y ) \, \, \text{(LC is orthogonal)} \\ &= -g(\nabla_{X}^{N}\xi, Y) \\ &= g(-(\nabla_{X}^{N}\xi)^{\top}, Y) \text{ (using that $Y$ is tangent)} \end{align}$ Thus $g\big( S_{\xi}X, \_ \big)=g(-\nabla_{X}^{N}\xi, \_)$, and applying $\sharp$ to both sides we get $S_{\xi}X=-(\nabla_{X}^{N}\xi)^{T}$ as desired. Plugging in $\nabla^{N}_{X}Y= \nabla_{X}^{M}Y+\textcolor{Thistle}{\mathrm{II}(X, Y)}$, $\begin{align} R^{N}(X,Y)Z&= \nabla_{X}^{N}\nabla_{Y}^{N} Z - \nabla_{Y}^{N}\nabla_{X}^{N}Z - \nabla_{[X,Y]}^{N}Z \,\, (\text{definition})\\ &= \nabla_{X}^{N} \big( \nabla_{Y}^{M}Z+\textcolor{Thistle}{\mathrm{II}(Y, Z)} \big) - \nabla^{N}_{Y} \big( \nabla_{X}^{M}Z+\textcolor{Thistle}{\mathrm{II}(X, Z)} \big) - \big(\nabla^{M}_{[X, Y]} Z + \textcolor{Thistle}{\mathrm{I I}([X, Y], Z)}\big) \\ &= \nabla_{X}^{M}\big( \nabla_{Y}^{M}Z+\textcolor{Thistle}{\mathrm{II}(Y, Z)} \big) + \textcolor{Thistle}{\mathrm{I I}\big(X, \nabla_{Y}^{M}Z+\textcolor{Thistle}{\mathrm{II}(Y, Z)} \big)} \\ & - \nabla_{Y}^{M} \big( \nabla_{X}^{M}Z+\textcolor{Thistle}{\mathrm{II}(X, Z)} \big) - \textcolor{Thistle}{\mathrm{II}\big( Y, \nabla_{X}^{M}Z+\textcolor{Thistle}{\mathrm{II}(X, Z)} \big)} \\ & - \big(\nabla^{M}_{[X, Y]} Z + \textcolor{Thistle}{\mathrm{I I}([X, Y], Z)}\big) \\ &= \end{align}$ [^7]: This is just first applying for a fixed [[vector field]] $X$ the [[musical isomorphism induced by a nondegenerate bilinear form|musical isomorphism]] to the covector field $h_{\xi}(X, \_)$, $S_{\xi}X:= \big( h_{\xi}(X, \_) \big)^{\sharp}$. Then varying $X$ gives a map $S_{\xi} \in \operatorname{End}(TM)$. ---- #### [^1]: Verifying this identification is an exercise (and includes verifying independence of choice of extension) [^2]: Verify indeed a symmetric 2-tensor (done in handwritten notes). The reason is essentially that $\nabla_{X}Y$ is not a tensor (it obeys Leibniz instead), and $\mathrm{II}$ is basically difference of two such partial covariant derivatives so that the non-tensorial components cancel out. Specifically using torsion-freeness have $\nabla^{N}_{X}Y-\nabla^{M}_{X}Y = [X,Y]-\nabla^{N}_{Y}X - [X,Y]^{T}+ \nabla_{Y}^{M}X=\operatorname{ I\! I}(Y,X)- \cancel{ [X,Y]^{\perp} }$ using that $X,Y$ are tangent to $M$. > [!specialization] Specializing to [[surface|surfaces]] in $\mathbb{R}^{3}$. > > > [!definition] Definition. ([[second fundamental form]]) > > For an [[orientation of a Euclidean submanifold|oriented]] [[differentiable Euclidean submanifold|surface]] $S$, the **second fundamental form** at $p \in S$ is the [[quadratic form]] $\begin{align} > \text{II}_{p} : & T_{p}S \times T_{p}S \to \mathbb{R} \\ > \text{II}_{p}(v,w)= & -dN_{p}(v) \cdot w, > \end{align}$ > where $N$ is the [[Gauss map]] and $dN_{p}=dN |_{p}$ is its [[differential of smooth maps between surfaces|differential]] evaluated at $p$. > \ > The [[matrix of a bilinear form|matrix]] of $\text{II}_{p}$ in the $\{ X_{u}, X_{v} \}$ [[basis]] of $T_{p}S$ is $\begin{bmatrix} > L & M \\ > M & N > \end{bmatrix}$ > where $L=\text{II}_{p}(X_{u},X_{v})$, $M=\text{II}_{p}(X_{u},X_{v}), N=\text{II}_{p}(X_{v}, X_{v})$. > > ^82a614 > > > [!basicproperties] Symmetry and Self-Adjointness. > > $\text{II}_{p}$ is a [[symmetric multilinear map|symmetric form]], i.e., for any two vectors $v,w \in T_{p}(S)$ $dN_{p}(w) \cdot v=dN_{p}(v) \cdot w.$ > > \ > > Fixing $w$, this also implies that $dN_{p}$ is [[self-adjoint]] as a [[linear operator]] on $\mathbb{R}^{2}$ (using that $T_{p}S \cong \mathbb{R}^{2}$ and $T_{N(p)}S \cong \mathbb{R}^{2}$). > > > > > [!basicexample] > > Suppose $X(u,v)$ is a [[coordinate patch|patch]] for a [[differentiable Euclidean submanifold|regular surface]] $S$ with $u>0$, $v>0$ and $X_{u}=(u,v,0), \ \ X_{v}=(0,u,0).$ > > > Let $p \in S$. Find the [[matrix]] of the [[second fundamental form]] under the [[basis]] $(X_{u}, X_{v})$. A normal vector is $\frac{(u,v,0) \times (0,u,0)}{\|(u,v,0) \times (0,u,0)\|_{2}}=\frac{(0,0,u^{2})}{u^{2}}=(0,0,1).$ > The [[Gauss map]] is then $N_{p} \equiv (0,0,1)$. So $dN_{p}$ is the zero map. Hence $\text{II}=\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$. > > ^477417 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```