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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

32835 questions
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Length of a curve is independent of parameterization

Suppose that $P:[a,b]\to \mathbb{R}^n$ and $Q:[c,d]\to \mathbb{R}^n$ be two parameterizations of the same continuously differentiable curve $\Gamma$. Can some one give a hint on how to prove that the length of the curve $\Gamma$ is independent of…
Kumara
  • 706
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Circle inscribed between two curves

Consider the plane region $S_n$ bounded from above and below for the graphs of $f_n(x)=x^{1/n}$ and $g_n(x)=x^n$, $0\le x\le1$. How to find the radious and center of the circle inscribed in $S_n$? Intuitively, the center is on the set $\{(A,A):0\le…
ksoriano
  • 141
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two points of the longest distance in a convex curve will be the vertices?

First we give the necessary definition: A vertex of a curve $\gamma(r)$ in $\mathbb{R}^2$ is a point where its signed curvature $\boldsymbol{k}_{s}$ has a stationary point, i.e.,where $d\boldsymbol{k}_{s}/dt=0$. We have Four Vertex Theorem: Every…
user39843
  • 937
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The Lie derivative does not determine a well-defined directional derivative

The errata for "Riemannian Manifolds: An Introduction to Curvature" by John Lee has for one of the problems the following correction below. Page 63, problem 4-3(b): Replace the first sentence by “Show that there are vector fields $V$ and $W$ on…
user782220
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Why doesn't ds appear in the statement of Green's Theorem?

I am trying to compare the line integral stated in Green's Theorem with the definition of a line integral. According to Wikipedia: $$ \oint_C(L dx+Mdy)=\int^b_af(\textbf{r}(t))|\textbf{r}'(t)|dt. $$ My intuition tells me that…
Tom O'Shaughnessy
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Find envelope of $x \sin \theta - y \cos \theta + z = a \theta$, where $\theta$ is a parameter.

I have tried to find envelope for $$x \sin \theta - y \cos \theta + z = a \theta$$ First I find derivative w.r.t. $\theta$ $$F(\theta)=x \sin \theta - y \cos \theta + z - a \theta = 0$$ $$\frac{\partial F(\theta)}{\partial \theta}=y \sin \theta + x…
Syed Muhammad Asad
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are geodesic shortest path or quickest path?

I'm a bit confused with geodesics : are they the shortest path (in distance) or the quickest path (in time). For example, Let take a triangle ABC. I'm using a car. I'm in $A$ and I have to go in $B$. The path $AB$ is 2km long, but I can go at 10…
Dylan
  • 465
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Definition of osculating plane

I couldn't understand the exact geometrical meaning of the osculating plane definition. Can any one help me with this? Thanks advance. Osculating plane: Let $\gamma$ be a smooth curve and P and Q be two neighboring points on $\gamma$. The limiting…
Selvakumar A
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How to compute curvature tensors for general n-dimensions?

I keep coming across calculations like this, Consider a metric on an $n+2$ dimensional manifold given as, $ds^2 = 2dudr + 2L(u,r)du^2 -r^2d\Omega_n^2$ Then apparently once can write down the Ricci and Einstein and other tensors as a function of n.…
Student
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A question on surfaces

If $S$ is the surface given by the function $z=y^2-x^2$, if I have the points $A=(1,0,-1)$, $B=(0,1,1)$, $C=(1,1,0)$, how can I use the Gaussian curvature to determine if there is an isometry of $S$ that takes $A$ into $B$? or $A$ into $C$? I know…
user62182
  • 370
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Orthogonality of principal curvature directions

I was trying to find a counterexample to the theorem that states that principal curvature directions are orthogonal. Obviously such an example doesn't exist but I'm having hard time understanding what's wrong with the following example. I construct…
Wazowski
  • 527
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Show that a specific map is a submersion of $O(3)$ in $S^2$

Denoting the components of the $3\times3$ matrix $A \in O(3)$ as $a_{ij}$, show that $$ F: O(3) \rightarrow S^2, a_{ij} \mapsto a_{1j} $$ is a submersion. (The map is well defined since for $A \in O(3)$ it is true that $a_{11}^2 + a_{12}^2 +…
user606572
  • 43
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Diffeomorphisms to $S^n$

Is $S^4$ diffeomorfhic to $S^2\times S^2$? Moreover. Is $S^n$ diffeomorphic to some cross product of manifolds $X\times Y$ for $n\geq2$? Is there a elemental topological invariant to let me see this? Any suggestions are welcome! thanks
Juan Carlos Castro
  • 329
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Why Hessian is a (0,2) symmetry tensor?

On a Remiannian manifold M, the Hessian of a smooth function $f$ on M is defined to be: $$\operatorname{Hess}f=\frac{1}{2}\mathcal L_{\nabla f}(g)$$ where $\mathcal L$ stands for Lie derivative, $g$ is the metric of the manifold, and $\nabla f$…
Yuyi Zhang
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Computing the curvature of a connection (a specific example)

I'm learning about the curvature of a vector bundle, and trying to compute it for a simple example. Let $\nabla$ be the connection on a trivial line bundle over $\mathbb{R}^2$ given by $\nabla s = ds$. I know that curvature is given by $F(X,Y)(s) =…
Anna
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