I need help with this differential geometry problem dealing with Gaussian curvature and mean curvature     Given an ellipsoid with parametric representation $(a\cos u\cos v,  b\cos u \sin v, c\sin u)$ compute its Gaussian curvature and mean   curvature.      (Hint: Compute $E,F,G$ first and $L, M, N$ second then Weingarten map   (Shape operator). The Gaussian curvature and the mean curvature are   the determinant and the trace of the Weingarten map (Shape operator).
123081
I am trying to review, and learn about how to compute and gaussian and mean curvature. Given $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$, how can I compute the gaussian and mean curvatures?  This is what I have so far, $$K(u, v) = \frac{a^2 b^2 c^2}{[c^2 \sin^2(v) (a^2 \sin^2(u)+b^2 \cos^2(u))+a^2 b^2 \cos^2(v)]^2}$$  Please help me out.
105737
Starting with these definitions  A curve on a manifold $\mathcal M$ is a smooth (i.e. $C^{\infty}$) map $\sigma $ from some open interval $(-\epsilon,\epsilon)$ of the real line into $\mathcal M$  Two curves $\sigma_1$ and $\sigma_2$ are tangent at a point $p$ in $\mathcal M$ if (a)  $\sigma_1(0) = \sigma_2(0) = p$ and (b) In some local coordinate system $(x^1,x^2,\ldots,x^m)$ around $p$, two curves are tangent in the usual sense as curves in $\mathbb R^m$, $$ (x^i \circ \sigma_1)'(0) = (x^i \circ \sigma_2)'(0)  $$ here, $i=1,\ldots,m$  The tangent vector is defined as the equivalence class of curves in $\mathcal M$ where the equivalence relation between two curves is that they are tangent at point $p$.  The tangent space is $T_p\mathcal M$ to $\mathcal M$ at point $p$ is the set of all tangent vectors at point $p$   I am trying to prove the tangent space at the point $p$ in a manifold $\mathcal M$ is a vector space. I'm starting with $v_1 \in T_p\mathcal M$, and $v_2 \in T_p\mathcal M$, and I have the following definitions $$ v_1 + v_2 := [\phi^{-1}\circ \ (\phi\ \circ \sigma_1 + \phi\ \circ \sigma_2 )] \\ r \ v_1 := [\phi^{-1}\circ \ (r  \phi\ \circ \sigma_1)]\ \forall r \in \mathbb R $$ I want to show that $v_1 + v_2 \in T_p \mathcal M$ and $r \ v_1 \in T_p \mathcal M$ As $v_1 ,v_2 \in T_p\mathcal M$, then $$ \sigma_1(0) = \sigma_2(0) = p $$ Now, for $v_1 + v_2$ be a vector at $p$ , $\phi^{-1}\circ \ (\phi\ \circ \sigma_1 + \phi\ \circ \sigma_2 )(0) = p$ $$ \phi^{-1}\circ \ (\phi\ \circ \sigma_1 + \phi\ \circ \sigma_2 )(0) = \phi^{-1} \ (\phi\ ( \sigma_1(0)) + \phi\ (\sigma_2(0)) ) \\ = \phi^{-1}((\phi\ ( p) + \phi\ (p) )) \\ =  \phi^{-1}( \  2\phi\ ( p) ) \neq p $$ I can't prove the closure relations starting from the definitions, what I'm doing wrong? Edit: The book I am following &quot;Isham, Chris J. Modern differential geometry for physicists. Vol. 61. World Scientific, 1999.&quot;, takes a special chart $(U,\phi)$ such that $\phi(p) = \mathbf 0 \in \mathcal M$, using this choice $$ \phi^{-1}\circ \ (\phi\ \circ \sigma_1 + \phi\ \circ \sigma_2 )(0) =  \phi^{-1}( \  2\phi\ ( p) ) = \phi^{-1}(0) = p $$ So, the closure is proven under addition. But this chart is a special choice. But the definitions hold for any charts around $p$, so another choice of charts should give the same result.
49714
I know what an ellipsoid is, but I do not know how to approach how to prove that one is convex. Any advice would be greatly appreciated.
69263
I am reading  by Bjørn Jahren, and have some questions about exercise 5.1.4, which says the following:     Let $\alpha(u)=(f(u),g(u)),u\in[a,b]$ be an embedded curve in the $xz$-plane such that $f (u) &gt; 0$ for all $u$. Find a parametrization of the surface of rotation obtained by rotating $\alpha$ around the $z$–axis and find conditions for this to be a regular surface.      Discuss what happens if we remove the condition $f (u) &gt; 0$.   A regular surface is defined by the following:     Definition 5.1.3. Let $S$ be a subset of $\mathbb{R}^3$ which is also a smooth surface, and let $\iota: S \to\mathbb{R}^3$ be the inclusion map. $S$ is called a regular surface if for every local parametrization $x$ of $S$ the Jacobian of the composition $\iota\circ x$ has rank 2 at every point.   A parametrization of this surface is given by $(u,t)\mapsto(f(u)\cos(t),f(u)\sin(t),g(u))$. By the definition of regular surfaces, we need the Jacobian of this parametrizationto be of rank 2. The Jacobian has the columns $(f'(u)\cos(t),f'(u)\sin(t),g'(u))$ and $(-f(u)\sin(t),f(u)\cos(t),0)$. If $g'(u)\neq0$ they are linearly independent if and only if $f(u)\neq 0$, since $\sin$ and $\cos$ are not 0 in the same point. If $g'(u)=0$ linear independence is equivalent to the matrix   $\begin{bmatrix} f'(u)\cos(t) &amp; -f(u)\sin(t)\\ f'(u)\sin(t) &amp; f(u)\cos(t) \end{bmatrix}$  having nonzero determinant, i.e. that $f(u)f'(u)\neq 0$. To conclude, the Jacobian has rank 2 if and only if $g'(u)\neq0\land f(u)\neq0$ or $f(u)\neq0\land f'(u)\neq 0$. Under the condition $f(u)&gt;0$, that is if and only if $\alpha'(u)\neq 0$.  Where do I go from here? I have found conditions given one specific parametrization, but I need to show it for all possible parametrizations. How do I generalize the result? ((EDIT: Maybe this suffices, since I can create an atlas with just this parametrization, by varying the domain?)) What about the points $a$ and $b$ are mapped to? Can I find a parametrization from an open subset of $\mathbb{R}^2$ around these? And what happens if we remove the condition $f(u)&gt;0$, how should I attack that?  Here is a handwritten (i.e. not so readable) and brief solution which may be of some use:  (Available .)
36416
Are there any geometric interpretation for the second partial derivative? i.e.  $$f_{xy} = \frac {\partial^2 f} {\partial x \partial y}$$  In particular, I'm trying to understand the determinant from  for determining whether a critical point is a minima/maxima/saddle points:  $$D(a, b) = f_{xx}(a,b) f_{yy}(a,b) - f_{xy}(a,b)^2$$  I have no trouble understanding $f_{xx}(x,y)$ and $f_{yy}(x,y)$ as the of measure of concavity/convexity of f in the direction of x and y axis. But what does $f_{xy}(x,y)$ means?
102047
Let $\mathbf{X} \sim \mathcal{N}_n( \mathbf{m}, \mathbf{C})$ be an $n$-dimensional gaussian vector, where $\mathbf{C} \in \mathbb{R}^{n \times n}$ is not diagonal, but it is positive-definitive, $\mathbf{C} \succ 0$, and $\mathbf{m} \neq \mathbf{0}$. Let $$ Y = \| \mathbf{X} \|^2 $$ where $\| \cdot \|$ denotes the $L_2$-norm (Euclidean norm), should follow a . Unfortunately, the Wikipedia page contains little information.  My question is: What is the PDF (probability density function) of $Y$ ?
114622
What is null geodesic equation for the static and spherically symmetric line element in $$ds^{2}=-K^{2}dt^{2}+\frac{dr^{2}}{K^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta{d\phi^{2}})$$ where $$K^{2}=1-\frac{2M}{r}-\Lambda r^{2}+2 a r$$ Does anyone know how to derive the null geodesic equation for the above metric? The result was given  without derivation as $$\frac{d^{2}u}{d\phi^{2}}+u=3M u^{2}-a$$ where $$u=\frac{1}{r}$$ I am trying to derive the above equation. First I started by setting $ds^{2}=0$, because null geodesic. Then I divided by $d\phi^{2}$. This is the equation I got $$\frac{dr}{d\phi}=\sqrt{K^{2}r^{2}\sin^{2}\theta}$$ Assuming that $\frac{dt}{d\phi}=\frac{d\theta}{d\phi}=0$, is this a correct assumption? Now Assuming that My calculation is correct so far does anyone know how to get the equation $$\frac{d^{2}u}{d\phi^{2}}+u=3M u^{2}-a$$
115349
I am using Do Carmo's Riemannian Geometry, and struggling to solve a problem.  The problem is:  Show that the mapping $F:\mathbb{R}^2\to\mathbb{R}^4$ given by  $$F(x,y)=((r\cos y+a)\cos x,(r\cos y+a)\sin x,r\sin y\cos\frac{x}{2},r\sin y\sin\frac{x}{2}))$$  induces an embedding of the Klein bottle into $\mathbb{R}^4$.  I know that the Klein bottle is defined as a quotient manifold $T^2/G$, where $T^2$ is the 2-torus, $G$ is a group of diffeomorphisms of $T^2$ formed by $\{A,Id\}$, and $A$ is the antipodal map, i.e. $A(p)=-p$.  Besides, I know that $T^2=S^1\times S^1$, where $S^1$ is the unit circle.  Also, I found that $F$ is injective.  However, I don't have a clue about how to attack this problem. I tried googling, but found nothing.  Any help will be appreciated.
95324
I was wondering if anyone could recommend some books for studying topics such as abstract manifolds, differential forms on manifolds, integration of differential forms, Stokes' theorem, de Rham cohomology, Hodge star operator? Our text is A Comprehensive Introduction to Differential Geometry by Spivak, but I think this book is very difficult for a beginner to learn.  Thanks in advance.
95287
In my class on differential geometry I have been given the following question on which I am stuck:  Let S be a regular orientable surface in $ R^3 $ with Gaussian curvature $K$ (not necessarily constant) and we are to check if there can be a smooth closed geodesic on S in the following case: $ K&gt;0 $ $ K=0 $ $ K&lt;0 $ and we are to give an example when it is possible that the closed geodesic bounds a simply connected region  I know for starters about the first one with positive curvature I may take the sphere and a great circle which is a geodesic and obviously bounds a simply connected subset of the sphere. But with the zero and negative curvatures I have no idea how to tell if a closed geodesic exists I think the Gauss Bonnet theorem has something to do with this but I cannot really proceed from there and I am posting here in the hope of getting help ******* Progress: I can work out what happens with Gauss Bonnet if I have the constraint that the geodesic bounds a simply connected region as this is simple but the fact of the matter is where I am stuck is the general existence or non existence if no constraint is given on the region bounded
70218
Show that if a curve $C ⊂ S$ is both a line of curvature and a geodesic, then $C$ is a plane curve.  Give an example of a line of curvature which is a plane curve and not a geodesic. (My thoughts: Take a sphere for $S$ and let the curve $C$ be any of the latitudes of $C$ that are not the equator or the poles.  Then $C$ will be a plane curve.  Additionally, since the curvature is constant on the whole of $S$, every direction is a principle direction (of sorts; this might be the flaw in my argument), so any curve is a line of curvature.  But, $C$ is not a segment of a great circle on $S$, so it is not a geodesic.)  Prove that a curve $C ⊂ S$ is both an asymptotic curve and a geodesic if and only if $C$ is a (segment of a) straight line.    Can I have some hints?  I am not very good at covariant derivatives yet, so hand-holding with detailed examples there would be nice.
30335
Let $M$ be a $2$-manifold (the setting I am wanted to apply this is $M=S^2$) and $g,h$ be conformally related metrics (ie $h=e^{2w}g$ for some $w\in C^\infty$). I would like to show that  $$ K_h=e^{-2w}(K_g-\Delta_g w) $$ where $K$ is the Gauss curvature. I have tried using the definition of Gauss curvature $$ K_g=\frac{\langle (\nabla_2 \nabla_1-\nabla_1 \nabla_2)e_1, e_2\rangle}{\text{det} g} $$ which introduces the factor $e^{-2w}$ (a factor of $e^{-4w}$ from the determinate and a factor of $e^{2w}$ from the inner product) but I do not see how the $-\Delta_g w$ term will appear. I also tried using normal coordinates but I still could not see how to get the $-\Delta_g w$ term. I was also reading  similar post but there was no answer to the question and I did not really understand what the comments were hinting at. I would really appreciate just a hint to get me started in the right direction.
119266
Consider the hyperbolic paraboloid $S$, $S=\{(x,y,z)\in\mathbb{R}^3 |z=x^2 -y^2\}$  Study the second fundamental form of this surface at $(0,0,0)$ and show that it is not a semi-definite bilinear form, In other words, show that its Gauss Curvature is negative at this point.  My solution:  First I define $f$ as $f(x,y,z)=-x^2+y^2+z$ and see that $\bigtriangledown f(x,y,z)=(-2x,2y,1)$ for all $(x,y,z)$ to define the Gauss map $N(p):S\rightarrow T_{p}S$ by   $N(p)=\displaystyle\frac{\bigtriangledown f(x,y,z)}{|\bigtriangledown f(x,y,z)|}$ $\forall p=(x,y,z)\in S$  where $T_{p}S$ is the tangent plane of $S$ at $p$  so, the Gauss curvature by definition is the product of two eigenvalues of  the differential $dN(p)$ that is given by the matrix  $DN_{(x,y,z)}=\frac{1}{|\bigtriangledown f(x,y,z)|}\left[\begin{array}{ccc} 2 &amp;  0  &amp; 0 \\ 0 &amp; -2  &amp; 0 \\ 0 &amp;  0  &amp; 0 \end{array} \right]$  and so $dN_{(x,y,z)}.(v_{1},v_{2},v_{3})=(-2v_{1},2v_{2},0)$, where $v\in T_{p}. S$  I checked that the eigenvalues of $DN$ are $\{2,-2,0\}$ but by the fact that the operator $dN(p)$ is self-adjoint it must have only two eigenvalues, so how can I choose these two? By the exercise, I assume that the curvature is $-2.2=-4$ but I don't know the argumentation for choosing these values.  I found an example  asking this same thing, but $S$ is a cylinder, in this case   $N(x,y,z)=\frac{1}{r}(x,y,0)$, $r$ is the radius of $S$ and  $dN_{(x,y,z)}(v_{1},v_{2},v_{3})=\frac{1}{r}(v_{1},v_{2},0)$  the matrix that defines $dN$ is   $DN=(1/r)\left[\begin{array}{ccc} 1 &amp;  0  &amp; 0 \\ 0 &amp; 1  &amp; 0 \\ 0 &amp;  0  &amp; 0 \end{array} \right]$  and the exercise concludes that the eigenvalues $k_{1}=1/r$ and $k_{2}=0$, in that way we can see there are only two options, but in my exercise there are 3.  I know this is kind of linear algebra exercise but i would appreciate any help of geometers and algebraists.
71315
I need your expertise in understanding and hopefully solving the following problem:  Let $C \subset \mathbb{R}^d$, and lets assume that $C$ is not centrally symmetric, thus when computing the $MVEE(C)$ (i.e the minimum volume enclosing ellipsoid), they use the following "lifting step":  $$ C^\prime = \left\lbrace \pm \begin{bmatrix} x\\ 1 \end{bmatrix} \middle| x \in C \right\rbrace,$$ where $C^\prime$ is centrally symmetric.  It is proven that for any $\varepsilon &gt; 0$, $MVEE(C^\prime) \cap \Pi$ is a $(1 + \varepsilon)$-approximation to $MVEE(C) \times \left\lbrace 1 \right\rbrace$, where $\Pi = \left\lbrace x \in \mathbb{R^{d+1}} \middle| x_{d+1} = 1 \right\rbrace$.  Let $G \in \mathbb{d \times d}$ be an invertible matrix, let $ v \in \mathbb{R}^d$ such that the pair $(G,v)$ represents the matrix and the center of $MVEE(C)$ respectively. Let $\hat{G} \in \mathbb{R}^{(d + 1) \times (d + 1)}$ be an invertible matrix such that the pair $(\hat{G},\overset{\to}{0})$ represents the matrix and center of $MVEE(C^\prime)$ respectively.  Then, $\forall x \in MVEE(C)$ $$ \left| \left| \left[ x , 1 \right] \hat{G} \begin{bmatrix} x\\1 \end{bmatrix} \right| \right|_2 = (1 + \varepsilon) \cdot \left| \left| (x - v) G (x - v)^T \right| \right|_2 .$$  Regarding $x \not\in MVEE(C) $, then it is easy to see that the above term is not satisfied and moreover the right term is bigger or equal to the left term.  What I need to know is whether it possible the bound the right term by the left term (maybe because $MVEE(C) \times \left\lbrace 1 \right\rbrace \subseteq MVEE(C^\prime)$), i.e. $$ poly(d) \cdot \left| \left| \left[ x , 1 \right] \hat{G} \begin{bmatrix} x\\1 \end{bmatrix} \right| \right|_2 \geq (1 + \varepsilon) \cdot \left| \left| (x - v) G (x - v)^T \right| \right|_2 \ ?$$  Please Do correct if I am wrong and thanks in advance.
1253
I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the circunference, then its curvature is constant.  I solved the second part since $\alpha$ is the circunference, then $\alpha(t)=(r\cos\theta,r\sin\theta)$, so $|\alpha'(t)|=r$, I need help in the first part.  Thanks a lot.
20170
I am looking for any help, hints, or suggestions in how to go about this problem from a previous qualifying exam.  We are given a smooth mapping $f: U \rightarrow \mathbb{R}^{n}$ whose differential $df_{p}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is of rank $n$ and we have that $\|f(x)-x\|&lt;\frac{1}{2}$. Here $U \subset \mathbb{R}^{n}$ is an open subset which contains the closed unit ball $B=B_{1}(0)$. The problem asks us to show that $\|f\|^{2}$ attains a minimum on the interior of  $B$.   As of right now, the only thing that comes to mind is the inverse function theorem, that there is an open subset $V \subset U$ where $f$ is an diffeomorphism on.
83520
I have just begun learning about the , , and  in the context of Riemannian manifolds.  However, I am finding it hard to relate these notions to any sort of geometric intuition.  How can one interpret these differential forms geometrically?  Or at least, what is the motivation for considering such objects?  What sort of information do they provide?  Edit: I should clarify that I think I have somewhat of an intuition for connections, the curvature tensor, and (to a lesser extent) the torsion tensor.  What I am asking about are the connection forms, torsion forms, and curvature forms.