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10

Let $S^n$ be a unit sphere in $\mathbb{R}^{n+m+2}$ realized in the first $n+1$ coordinates and $S^m$ be similarly in the last $m+1$ coordinates. Then the distance of any $(s_1,0)$ to $(0,s_2)$ is $$ s_1^2+s_2^2=2. $$ I doubt that you can realize such spheres in smaller dimension $<m+n+2$, but don't see how to prove you can't.


8

I think that it is possible to deform a curve in a way that we can vary torsion while keeping curvature fixed. I was wondering if anybody could give me a nice example of this. Arguably the simplest class of examples are those for which the curvature $\kappa$ and torsion $\tau$ are constants for each curve in the family. Any such curve is a helix ...


7

The fact is that the objects of study in differential geometry (smooth manifolds) build on and generalize the objects of study in elementary geometry (lines, circles, spheres, cylinders, etc.) much more than they generalize the objects of calculus. It's the tools of calculus that are directly generalized to create the fundamental tools of differential ...


6

Since you can transport along the curve in the inverse direction, and the composition of the two transports is the identity map, yes. This follows at once from the uniqueness of solutions of differential equations with initial conditions.


5

This question perhaps risks closure on account of the (in my view) enormous breadth of the main question, but I'll try to answer the specific questions asked afterward. I'll assume "introductory calculus" simply means single- and multivariable differential and integral calculus. An analogue of a differentiable manifold is (very roughly) anything one ...


5

The claim is not that the (surface) area, $4 \pi r^2$, of a sphere of radius $r$ grows more slowly than the area, $\pi r^2$, of a disk of radius $r$ on the plane. Rather, the book is asserting that the area of a disk on a sphere grows more slowly with radius than does a disk on the plane. If we pick a point $P$ on a sphere of radius $R$, we can ask which ...


3

In fact, I'd prefer the second definition - it's what Lee uses in his Riemannian Manifolds book: This allows a curve to intersect itself at a point in different directions while being able to record this fact in the velocity vector. I think if one takes seriously the idea that "curve" refers to the map $\gamma:I\to M$, and not its image ...


3

Yes, absolutely. The approach I know about is synthetic differential geometry, which begins with thinking of manifolds not via their underlying locally Euclidean topology (this is analytic differential geometry!) but via their smooth functions, in particular those to and from the line $R$ ($R$ behaves differently enough from $\mathbb{R}$ that it's a good ...


3

Your formulas are fine. The "coordinate representation" of a map from $\mathbb R$ into a manifold is generally defined only on a subset of $\mathbb R$, namely the subset that maps into the domain of the coordinates. In this case, the domain of stereographic coordinates is $U = \mathbb S^3\smallsetminus \{(0,0,0,1)\}$. For any $z$ of the form $z=(0,z^2)$ (or ...


2

Any two Riemannian manifolds with constant sectional curvature $C$ are locally homogeneous (in normal coordinates, one has an explicit description of the metric and by composing two normal coordinate systems around two different points one obtains a local isometry). However, such spaces need not be homogeneous. For example, consider a closed oriented ...


2

For the first question: Correct -- a one-dimensional space cannot have any intrinsic curvature; the Riemann tensor always vanishes. For the second: In 3-dimensional hyperbolic space, the Gaussian curvature of a geodesic plane will be $-1$, because it it itself a 2-dimensional hyperbolic space. This is most easily seen by considering the hyperplane that is ...


2

I am not sure I understand your question, but the canonical divisor of $C\subset \mathbb P^2$ is $C\cdot E$, with $E$ as you wrote a general curve of degree $d-3$. At the level of invertible sheaves: $$K_C=(K_{\mathbb P^2}+C)|_C=\mathcal O_{\mathbb P^2}(-3H+C)|_C=\mathcal O_C(d-3).$$ The key word here is adjunction formula.


2

I think you're correct about the typo. Perhaps he meant $X_{11} \cdot X_1 = \frac{1}{2} (\Lambda^2)_1$ or something. If you use your corrections, then the expression just falls out with a bit of writing. We calculate $$E_1 = \alpha_1 X_1 + \alpha X_{11} + \beta_1 X_2 + \beta X_{21} + \gamma_1 N + \gamma N_1$$ $$E_2 = \alpha_2 X_1 + \alpha X_{12} + \beta_2 ...


2

What is "the" volume form? If you work with the induced metric on $S^{n-1}$ from $\mathbb{R}^n$, then the map $A$ is an isometry (assuming $S^{n-1}$ is centered at the origin) and in particular, preserves the Riemannian volume form induced by the metric.


2

Technically, $g_{1}$ is a metric on the unit sphere $S^{n-1}(1)$ and $g_{r}$ is a metric on the sphere $S^{n-1}(r)$ of radius $r$, so they're not immediately comparable. What you're proven is: If $\phi:S^{n-1}(1) \to S^{n-1}(r)$ is radial scaling by a factor of $r$ (i.e., $\phi(x) = rx$), and if $g_{r}$ is the metric induced on $S^{n-1}(r)$ by the ambient ...


2

Let $S$ be a regular surface, $f$ a local parametrization in some neighborhood of a point $p$, and $f_{x}$, $f_{y}$ the coordinate vector fields, and $N$ a continuous unit normal field. Notation below follows the exam mentioned in the comments. The first fundamental form is the quadratic form on $T_{p}S$ whose matrix with respect to the basis $\{f_{x}, ...


2

The theorem you cite has a weaker version which makes no claim about the boundary of $X$. Theorem: Suppose that $F:X\times S\to Y$ is a smooth map of manifolds and $Z$ is a submanifold of $Y$, all manifolds without boundary. If $F$ is transverse to $Z$ then for almost every $s\in S$ the map $f_s : x\mapsto F(x,s)$ is transverse to $Z$. We can deduce ...


2

For completess, here is how the rest goes. The radial part of the Euclidean Laplacian (without minus sign) in $\mathbb{R}^{n+1}$ is $r^{-n}(r^n u_r)_r$ (derived here). If $u$ is a homogeneous polynomial of degree $k$, then $u(r\xi)=r^k u(\xi)$ (with $\xi\in S^n$), hence $$r^{-n}(r^n u_r)_r =- r^{-n}(k r^{n+k-1})' u(\xi) = k(n+k-1)r^{k-2}u(\xi)$$ Since $u$ ...


2

When $n>1$, $S^n$ is simply connected, and $H^1_{dR}(M) = 0$ for any simply connected manifold. (Hint: For any closed $1$-form $\omega$ and any closed curve $\gamma$, we have $\displaystyle\int_\gamma\omega = 0$.) Now, in order to deal with $\Bbb RP^n$, consider the $\Bbb Z/2\Bbb Z$ action given by the deck transformations. Show that any exact invariant ...


2

The problem of to show that the surface with minimal area that enclose a given volume is a sphere is not at all simple. Using the calculus of variation we can show that the surface must have constant curvature. I sketch the proof: Given a smooth closed surface $S$ in $\mathbb{R}^3$, let $A$ its area and $V$ the volume enclosed. As varied surface consider ...


2

If I am not mistaking, the statement Sphere is the only closed surface in $\Bbb R^3$ that minimizes the surface area to volume ratio. is equivalent to either of the following statements: Sphere is the only closed surface in $\Bbb R^3$ that minimizes surface area, enclosed volume fixed. Sphere is the only closed surface in $\Bbb R^3$ that maximizes ...


2

Hint Suppose $\Omega^m$ is exact; what does Stokes' Theorem say about the integral $\int_M \Omega^m$?


2

$\bullet \space \mathbf{SO(3) / SO(2) \simeq S^2}:$ Consider a fundamental representation of the Lie group $G := SO(3)$. Any element $M$ of $G$ can be written as a linear map $M : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $M^{-1} = M^T$ and $\det(M) = 1$. We can easily restrict to $M : S^2 \rightarrow S^2$. For any arbitrary $x \in S^2 \subset ...


2

For $M=\mathbb R^n$ take a smooth function with compact support $\phi\in C_0^\infty(M)$. Then $$ \phi = (-\Delta+\lambda I)^{-1}((-\Delta+\lambda I)\phi) $$ obviously. Now consider translations $\phi_k(x):=\phi(x+kv)$ for $v\in \mathbb R^n$, $k\in \mathbb N$. Take the norm of $v$ large enough such that the supports of different $\phi_k$'s are empty. The ...


2

If I haven't missed something, the only thing you cannot do in the case of (a) is adding $M\setminus S$ to the covering. Despite this, $\{W_p\mid p\in S\}$ is still a covering of some neighborhood $$U=\bigcup_{p\in S}W_p$$ of $S$. Therefore, simply taking a partition of unity $(\phi_p)_{p\in S}$ subordinate to this cover allows you to construct the desired ...


2

No, consider $p(re^{it}) = (\sin r )e^{it}.$ Then $p$ maps $K = \overline {D(0,3\pi /4)}$ onto $\overline {D(0,1)}$ but $p(\partial K)$ doesn't even intersect $\partial (p(K)).$


2

You have many questions in one1), but let me only calculate the formal adjoint. In the Sobolev space $H^k=W^{k,2}(\mathbb R^n)$, $k\geq1$, one can use the inner product $$ \langle u,v\rangle_{H^k} = \sum_{|\alpha|\leq k}\langle\partial^\alpha u,\partial^\alpha v\rangle_{L^2}. $$ Compactly supported smooth functions are dense, so let us work with $u,v\in ...


1

There are many. The simplest is Cayley's transformation, which is defined by: $$z\mapsto\frac{z-\mathrm i}{z+\mathrm i}.$$ More generally you have the Möbius transformation: $$z\mapsto\mathrm e^{\mathrm i\mkern1.5mu\theta}\frac{z-z_0}{z-\overline z_0}$$ which sends $z_0$ to the centre of the disc.


1

It's pretty much a classical result that the catenoid and helicoid are isometric; if you look up a proof of that, you'll probably be able to discover the inconsistency in your reasoning. One thought: there's no reason that the isometry has to take $x$ to $x$ and $y$ to $y$ in your chosen parameterizations, so your reasoning about the coefficients of dx and ...



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