Interpolating a "manifold" between two points - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:05:57Z http://mathoverflow.net/feeds/question/109643 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109643/interpolating-a-manifold-between-two-points Interpolating a "manifold" between two points Chirag Lakhani 2012-10-14T19:35:08Z 2012-10-16T15:12:25Z <p>Edit: I have reworded the question.</p> <p>This may be a basic question but I am having trouble figuring out the correct answer. I want to find a local coordinate chart that fits a d-dimensional submanifold in $\mathbb{R}^N$. I am given two points $p_1, p_2 \in \mathbb{R}^N$ and corresponding orthonormal bases $(\phi_1, \phi_2, \ldots, \phi_d)$, $(\tau_1, \tau_2, \ldots, \tau_d) \subseteq \mathbb{R}^N$ for their tangent spaces. I would like to find an algorithmic method for finding functions $(f_1(x_1,x_2,\ldots,x_d), f_2(x_1,x_2,\ldots,x_d),\ldots f_N(x_1,x_2,\ldots,x_d))$ that satisfy these conditions. </p> http://mathoverflow.net/questions/109643/interpolating-a-manifold-between-two-points/109676#109676 Answer by Ryan Budney for Interpolating a "manifold" between two points Ryan Budney 2012-10-15T03:32:24Z 2012-10-15T03:40:55Z <p>There isn't a formula for what you're looking for. At least, the formula can't make sense for all initial data <em>and</em> depend continuously on that initial data. </p> <p>Your initial data is two points in the Stiefel manifold $V_{n,d}$, together with two points in $\mathbb R^n$. If you were to find an embedding $B^d \to \mathbb R^n$ whose derivatives agreed at two points with these points of your Stiefel space, you'd have produced a section of the fibration:</p> <p>$$Emb(B^d, \mathbb R^n) \to V_{n,d}^2 \times C_2 \mathbb R^n$$</p> <p>where this map is given by restriction to the endpoints. $C_2 \mathbb R^n$ is the configuration space of two points in $\mathbb R^n$, it has the homotopy-type of a sphere. </p> <p>$Emb(B^d, \mathbb R^n)$ has the homotopy-type of $V_{n,d}$. So if your section existed, the homotopy and homology groups of $V_{n,d}^2 \times C_2 \mathbb R^n$ would embed in the homotopy and homology groups of $V_{n,d}$. Since the first non-trivial homotopy group of $V_{n,d}$ is cyclic, this is impossible. </p> <p>Admittedly, I'm making some extra assumptions that you did not specify, but I think this shows you that whatever your solution is, it has to be somewhat nuanced. </p> http://mathoverflow.net/questions/109643/interpolating-a-manifold-between-two-points/109824#109824 Answer by Brendan Foreman for Interpolating a "manifold" between two points Brendan Foreman 2012-10-16T15:12:25Z 2012-10-16T15:12:25Z <p>This isn't really a formula so much as a conceptual construction.</p> <p>First suppose that $d=n-1.$ </p> <p>Let $T_1=span(\phi_1,..., \phi_{n-1})$ and $T_2=span(\tau_1,...,\tau_{n-1})$ be the tangent subspaces of $T{\mathbb R}^N$ at $p_1$ and $p_2,$ respectively. We can think of these as hyperplanes in real N-space so that there is an intersection of half-spaces created by these hyperplanes containing the line segment $l=p_1 p_2.$ Choose $r>0$ small enough that the spheres tangent to $T_1$ and $T_2$ at $p_1$ and $p_2$ with radii equal to $r$ and centers located on $l$ do not intersect. Call these spheres $S_1$ and $S_2$. Define a path of spheres by $S_{1+\lambda}=(1-\lambda)S_1 + \lambda S_2$ for $0\leq \lambda \leq 1,$ where addition is given by Minkowski addition of subsets in $N$-space.</p> <p>The envelope of this path of spheres should be the boundary of a tubular neighborhood of a line segment, which is tangent to $T_1$ and $T_2.$</p> <p>Second suppose that $d$ is strictly less than $n-1.$ Fill in the given orthonormal bases so that we have the above case and make the above construction. Then we let $T_1$ and $T_2$ be the subspaces at $p_1$ and $p_2$ spanned by the original given orthonormal bases and set $T_{1+\lambda}=(1-\lambda)T_1 + \lambda T_2$. The intersection of $T_{1+\lambda}$ with the corresponding $(1+\lambda)$-element from the envelope as constructed above should determine a $d$-dimensional submanifold satisfying the desired conditions.</p>